The problem is, almost every single way in which 5e is "like" 4e has some critical change that...rather weakens the direct effects of the thing in question. That is, it's reasonable to say that there are several 5e mechanics that hold a vague resemblance to 4e mechanics...but it's (IMO, anyway) nearly false to say that 5e uses a lot of 4e mechanics. Allow me to explain, with some illustrative examples of where 5e did, and did not, keep the spirit of a 4e mechanic, or merely the most distantly abstracted mechanical concept of 4e.
Actually similar: 4e Backgrounds+Themes vs 5e Backgrounds
While IMO 4e Backgrounds+Themes are overall better even when compared to 5e in-context, as there's more of the former to provide more variety and yet they're simultaneously more AND less mechanically weighty,* 5e Backgrounds pretty well cover the conceptual space 4e did with these two mechanics combined. It isn't a purely high-level seeming; the two editions share basically the same spirit here, 5e just simplifies them because it made simplification/"streamlining" a central goal, and that's perfectly fine. Some of the later, setting-specific 5e Backgrounds, such as the Ravnica options, are even more similar to 4e Themes, providing additional benefits as you gain levels (extra spells) and otherwise doing more than just "1st-level feature and some skills."
(*4e BGs are very simple mechanically, even by 5e standards. 4e Themes are not so simple, but you never need to take anything from them other than the 1st-level feature, making them theoretically no more weighty than any 5e option. Hence, in a weird way, 4e BG+Theme is both more and less weighty. It's an odd situation.)
Vaguely similar, but shorn of critical parts: Healing Surges vs Hit Dice
People often get confused about why some 4e fans, including myself, really really dislike Hit Dice despite really liking Healing Surges. "Aren't they the same thing?" is a (paraphrased) question I've heard several times. And the answer is they are, and are not, "the same"--but the "are not" is more weighty here.
At the highest-level abstraction, "a personal resource that lets a character heal," Hit Dice and Healing Surges perform the same function, just with different amounts. Since "different amounts" is true of nearly every comparison between editions, even 1e to 2e just to a smaller degree than most edition changes, if this were the only way we could compare HD and HS, I would gladly call them functionally equivalent. But that's not the only way to compare them, and the part that got cut is where the vital difference lies.
See, Healing Surges aren't just "this resource is for healing." They served an additional, vital function: Capping daily healing. Yes, there were a few ways to skirt around this limit a little bit, but those were always in short supply as well. Either they were daily powers (and thus effectively "this power grants an extra surge or two once a day"), or they were addressed by the Bag of Rats rule (no beating up a bag of rats to get infinite HP). Further, almost all healing other than the aforementioned daily healing abilities (like Clerics with the "Cure X Wounds" series) required healing surges to function--even healing potions did! This meant everyone, even a Slayer or some other (sub)class that didn't give two figs about daily powers, had a metaphorical "clock" running down until they had to rest. That 5e removed this is a really, really big difference; HD are a nicety added on top of the critical necessity for magical healing, while HS are the core of the healing system and magic can do little to stretch things any further than surges allow.
I will, however, note at least one way in which surges and HD can be used similarly: Justifying Warlord healing. Since the vast majority of healing in 4e depended on the recipient spending a surge, it was perfectly reasonable for a non-magical class to have the ability to coax, command, or compel an ally to draw on that inner strength. HD can be used for exactly the same purpose (and, for a handful of mechanics, already have been used for that purpose). So I don't mean to say that there's ABSOLUTELY NOTHING of the "spirit" of the 4e Healing Surge rules in Hit Dice. But the key conceptual component, that actually provided incentives for player behavior and discouraged 3e-style "just use a wand of CLW, 4head" play, has been clipped out entirely, and that severely dampens the enthusiasm a 4e player might have for HD.
Actually similar: Feats
You already covered this, it sounds like, so I'll keep it brief. Feats are, by and large, the same mechanic working in the same way. Some of them work very differently, but the spirit of the rule is the same, and arguably more similar to 4e than 3e. All those "pick up a feature from class x" feats, for example, are much more similar to 4e's multiclass feats than they are to anything from 3e. I know this is going to look small by comparison to the other two, 'cause I'm not doing any analysis here, but it really bears repeating: feats are arguably the MOST well-translated part of 4e that appears in 5e. They don't work the same--they shouldn't, the two games work differently--but the spirit of the rule is as close to identical as two differently-structured games can be, IMO.
Vaguely similar, but shorn of critical parts: Half-Level Bonus vs Proficiency
This one's pretty simple, or so you'd think. On the similarity front, it's very direct: Proficiency scales at "quarter-level rounded up, plus one," so it's (very nearly exactly) half the half-level bonus of 4e. That's a cut and dried similarity, right? Surely this must mean that 5e kept the spirit of the 4e rule if it is mathematically identical apart from cutting the value in half?
Well....no. Because the critical conceptual bit of 4e's half-level bonus is that it applies to everything. Now, obviously, this was (and remains) somewhat controversial; some players find it utterly baffling that a character could just "get better" at any skill whatsoever, while others felt they were "on a treadmill." (I have major issues with that way of describing how 4e works, but that's a separate topic.) But we've already seen some of the ways that 5e's method can come up short: many fans, at least for the first few years, grumbled quite a lot about how having only two saves that scale means a character, pretty much axiomatically, falls behind on every other saving throw--potentially including at least one "important" saving throw (Con, Dex, or Wis).
Further, with the limited number of skills available to a character (unless they drop feats on getting more, which is its own controversy), if a skill is neither one related to a primary attribute nor one you're proficient with, you're never going to get any better with it, but the world will throw progressively harder challenges at you--aka "treadmill," even if a shallower one. Monster Perception scores, for example, scale up with CR. If you have Dex 8 and no Stealth proficiency (and doubly so if you wear armor that imposes Disadvantage on Stealth), not only will you be bad at Stealth to start with, you will slowly lose ground as the game progresses. You don't just have a weakness; you have a permanent fault that, unless you blow resources on getting better at it, you will fall further behind with.
Now, this is not to say that 4e was some paradise where everyone got everything they wanted always. You did still see higher perception scores in higher-level monsters, and if you find a lock in the dungeons of a mad demigod, it's probably going to have a higher DC than a lock you'll find in a crooked merchant's basement. The critical thing, though, is that a level 20 Paladin, even one who dumped Dex and never took Stealth proficiency, is more capable of sneaking past an ordinary guard, should such a situation arise. They've learned a thing or two by adventuring. Will they be able to avoid detection from the living shades that guard the Transcendent Soulmaster's lair in the Negative Energy Plane? Probably not! But if they go back and check out lower-level threats, the Paladin does relatively improve. That sort of thing is impossible in 5e; an equivalent Paladin never gets any better at sneaking past goblin mooks, no matter how many levels she gains.
Vaguely similar, but shorn of critical parts: At-Wills vs Cantrips
This one is a very, very common assertion of how 4e lives on in 5e, and it's very frustrating as a result. Because, on the one hand, yes: at the highest-level abstraction, considering no other context than "what does X thing do?", at-wills and cantrips are literally identical--some cantrips even originated in 4e (such as vicious mockery). But if you consider even a tiny bit more detail than that, the break becomes readily apparent.
See...cantrips only benefit casters. That's huge. 4e tried to get away from benefits that applied only to casters. That's part of why it had non-AC Defenses instead of Saves (more on this later), and Healing Surges, and Implements, and the Warlord, and several other things besides. At-wills meant that characters who weren't overtly "magical" in nature still had some neat trick they could pull out on the regular, often something that supported a particular class fantasy or behavior. For example, the Brawler Fighter wanted Grappling Strike, which fed into the class fantasy of being an unarmed (or weapon-and-fist) character. The Warlord could take a lazy at-will, or something more proactive, depending on the style of leadership they wanted to support. Etc.
Having characters that just make more attack rolls doesn't fit that. Having Battle Masters who only get 4-6 special attacks every 2 fights (on average, as intended by the design team) doesn't fit that. Yes, some of the stuff that was once provided by at-wills is now doable generically (e.g. targeting more than one opponent or shoving or whatever) by expending individual attacks within Extra Attack to do it. But cantrips are the only part of 5e that retains the full diversity and utility of at-wills, and it gives those features only to casters. That's directly contrary to the spirit of 4e, and a major reason why this is a sore spot for big fans thereof (such as myself).
Mostly similar...depending on DM: Skills
It's pretty much inarguable that 5e used a skill system more similar to 4e than 3e. There are fewer skills overall than 3e, you only pick specific skills to be good with at character creation and only get more by expending resources (primarily multiclassing and feats), training is a "you have it or you don't" situation, super-training is a similar "you have it or you don't" situation (4e just includes more granularity within it), etc. I don't think anyone can argue that 5e's skill system doesn't obviously crib from 4e's.
The big problem, I find, is in the ways DMs use it. My experience of 5e skills in play resembles my experience of 3e skills more than 4e ones. 4e skills were explicitly meant to be very, very broad. Arcana was "is it magic, and not god-magic or nature-magic? Then you have a chance to know/learn something about it." Nature, likewise, applied to basically anything in the natural world, and most things in the Primal power source, too. Streetwise covered basically anything social that wasn't "tell a lie," "ask nicely," or "ask nastily"--casing the joint, scrounging for rumors, finding a fence, local events, sizing up a crowd, etc.
My experience with 5e, and I do want to emphasize that this is a personal thing, is that skills are viewed more as narrow things (the way 3e approached them) rather than broad ones (the way 4e did). I've no good explanation for why this is the case, but it is what I've seen. Perhaps the fine splitting between "Investigation" and "Perception," or pulling "Animal Handling" out of "Nature," I don't know. But it's what I've seen.
Vaguely similar, but shorn of critical parts: Magic Items
Magic items have been a hot-button topic since at least the days of 3.5e. 3rd edition had secret expectations of magic items, that only got called out officially in semi-fluff stuff (like the "X With Class" series)--you needed AC boosters, saving throw items, magic weapons if you made attacks, that sort of thing. 4e was honest enough to make these expectations explicit, and structured the rules so that the benefit of magic items was clearly defined and generally expected. Players, in general, want cool magic toys, so it seemed reasonable to make a system where you get magic toys as part of play, rather than trying to obfuscate their presence behind the curtain. These items ran from +1 to +6 in power over the course of 30 levels, effectively +1 per five character levels. (This even got codified in the fan-named "automatic bonus progression" rules, which were a big hit and quite popular in settings like Dark Sun where magic weapons are rare.)
Some people absolutely hated this idea, and derided 4e as having "magic Wal-mart" or being a "Christmas tree" of loot. It was basically the Monty Haul edition-war concept, just rephrased for the modern, discerning edition-warrior. So, when 5e came along, this obviously had to be dealt with! So they told people magic items were totally, completely optional. Exceeeeept...that's not really true. I mean, it's theoretically true in that, if you're selective enough about what monsters you field, there is in principle no absolute requirement that PCs have magic items. But we have, absolutely, gone back to the "hidden behind the curtain" way of doing things. There's a significant number of monsters--I don't know the statistics, but it's definitely at least a large minority--that are much more difficult to harm if you don't have magic weapons. When coupled with the "HP is the main metric of scaling, not defenses" philosophy of 5e, this means magic items are soft-required. Sure, you don't HAVE to have them....but if you DON'T have them, many enemies are going to become extremely un-fun slogs, unless the party spellcasters take care of it for you. (Yet another "the casters have benefits and the non-casters don't" thing.)
Some will note, relating to the above half-level vs. PB thing above, that 5e weapons and armor have exactly half the scaling 4e ones did, +1 to +3 rather than +1 to +6. This is fair, but...well, because magic weapons and armor were so stridently called optional (regardless of the practicalities of that statement), my experience is that many 5e DMs see them as blatant power creep, and thus hand out few if any to the players. It is an irony of the "Bounded Accuracy" system (which, as I've said elsewhere, is neither all that much concerned with accuracy, nor with boundedness!) that in trying to make magic items feel like something you can use freely because they aren't expected, it has instead made them seem to a great many people like they're inappropriate to hand out without extra-special justification.
Intentionally not similar: Defenses vs. Saves
This one's gonna be controversial, in part because it's very clearly a "going back to 3e's way of doing things" area. D&D, prior to 4e, mostly used saving throws for magic and attack rolls for physical hits. There were exceptions, so anyone claiming that the two mechanics cleanly distinguished spells from non-spells (or even magic from non-magic) is simply wrong, but in general if it wasn't a magical offensive effect, it resolved with an attack roll, and if it was a magical offensive effect, it resolved with a saving throw. 5e, obviously, has played with the formula a little bit, but by and large they kept things basically the same, just making all six stats (theoretically) relevant, rather than only three.
Thing is, there was an extremely good reason why 4e went to "NADs" (as fans semi-affectionately called them) rather than saving throws: very simply, was making support-focused characters more useful. Consider a very simple situation: Bard that favors supportive magic is playing alongside both a Sorcerer and a Rogue. In 5e, the Bard has to choose which person to help, because (most of) the Sorcerer's kit depends on enemies failing saving throws for its best damage, while (most of) the Rogue's kit depends on being accurate with attacks. The Bard basically cannot support both characters equally; they can either do stuff to boost the Rogue's attacks, or stuff to weaken a target's saving throws, but can't really ever support both of their friends at the same time. In 4e, however, any support power (whether it comes from a leader or not) that increases hit can help anyone in the group. Further, if an effect improves all nearby allies, everyone gets equal benefit--doesn't matter if you're a Wizard or a Fighter, +1 attack is +1 attack. And since NADs scale a little differently from AC, debuffing an enemy's Reflex defense is not strictly beneficial to the magic-users in the party--plenty of Rogue, Ranger, and other Martial classes' attacks target Reflex (or Fortitude; few target Will, but IIRC some do exist, mostly in the realm of "being an intimidating combatant" aka similar to an early-edition Morale check).
So...yeah. There are probably more areas I could mention, but this post is already super long and I'm drawing a blank as to more things I could mention. It is absolutely the case that you can find parallels between 4e and 5e, but whether those parallels are merely skin-deep or actually serious varies wildly from one mechanic to another.
Yeah....see above about that. This glosses over an extremely important difference between 4e's half-level bonus and 5e's Proficiency.
