There are many differences and many similarities. I think if you understood how any of the d20 system worked, you will be able to understand both systems, though as always, the devil is in the detail.
Basically, the core idea is still this:
Roll a d20, preferably high, to resolve any game interaction.
Characters have classes.
Classes have attacks, defensive values (AC, Reflex, Will, Fortitude), skills and feats and hit points. Also, there are some class-specific aspects for each character, and here you probably find the biggest variety, often related to resource management.
3E approach to the "class-specific" aspects is to often introduce class-specific subsystems.
Examples:
the Barbarians class specific subsystems is Rage, and to a lesser extent Uncanny Dodge, Trap Sense and Damage Reduction.
Rogue has Sneak Attack, later some Rogue feat-like abilities, and Uncanny Dodge.
Rangers have spells and combat styles.
Fighters have bonus feats.
Wizards have spells (of the arcana variant)
Sorcerers also have spells, but use a different way to resolve spell
Each of these class-specific subsystems makes the entire game experience for each class very unique, and it also tends to reward a certain "rules mastery" for each class - you might love playing Wizards, but have trouble playing Clerics or Rogues, until you spend a similar amount of time learning the fineprints of their subsystems.
Pathfinder continues to go along these lines. It makes the sub-systems more intricate - Barbarians get Rage Points to fuel special abilities with Rage, Monk gets Ki Points for similar purposes, Paladins get Aura abilities and so on.
4E goes another route. It introduces a basic "power" system for every class. The structure is identical, so if you switch the class, you don't need to learn a new way to manage your resources. The powers are described from simple building blocks. Think of the start of each 3E spell description (the list of level, range, saving throw, spell resistance, duration, casting time), but shorter. The mechanical terms that describe what a power does are also well-defined and allow very short and understandable descriptions of powers. Keywords are - well, key.
It doesn't take long to know what a spell or a martial maneuver (exploit) does.
The devil is not in the detail, but in the composition - how do you use your own powers and that of your comrades in an encounter. Understanding the individual power might be easy, but how you use it with most tactical advantage is difficult.
Of course, the introduction of more group tactics in 4E doesn't mean it didn't exist previously. But there seem to be more effects relying on this. For example, a solid tactic for Rogues in 3E was to get into flanking position - no surprise. It's still a solid tactic in 4E. But 4E adds more situation where a Rogue can try to get the required "combat advantage" against his foes, since powers can add such conditions to enemies, making the Rogue more flexible on the battle-field - if the party figures out how to use these powers to best advantage and against the right targets.
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Another difference might be the so called difference along the "GNS" scheme (sometimes controversely discussed on EN World").
4E is basically a more Gamist and slightly more Narravist system, at the expense of Simulation.
3E is also Gamist in many regards (the whole notion of levels and hit points alone, but also in the notions of class balance.), but has a stronger "simulationist" streak.
Example for "Gamist" vs "Simulationist":
Monsters and NPCs in 3E follow the same rules for player characters. "Pure" monsters without levels don't use class levels, but type hd (which might be not entirely consistent from some point of view). The HD by type link a types attacks, saves and hit points and in a certain way is supposed to "simulate" the difference between an animal-like creature and a dragon-like or an undead creature.
4E gives up the notion of tying these attributes to type. While type and subtypes still exist, monster abilities are defined by their metagame role in a combat encounter. A monster that is to use a lot of ranged attacks uses hit points by level for an Artillery Monster, while a monster that strikes from hiding is a Lurker. An Orc (as a special type of monster) is not automatically a Brute or a Soldier, but he can be any of it, depending on what is purpose is in combat.
The underlying philosophy here is more "Gamist", because it focuses on the usability aspect when playing the game, as opposed to simulate how such a creature might actually "grow" up.
Using the type HD in 3E, you could easily imagine that a monster "grows" these HD with age, and they really represent some physical qualities. Dragons are tough beasts, therefore they get d12. Feys have a weak and soft physique, therefore they get d6. Undeads are typically mindless and unskilled, but tough, therefore they get d12s but a low BAB, and so on.
An example of "Narrative" vs "Simulation" is the power systems.
3E assumes that martial abilities are "mundane", and therefore are basically repeatable, even if they might require some setup (you can cleave only if you kill a foe, but you can surely do it all the time). Magic, being outside of our realm of experience, can use any arbitrary system to be resolved, and still be considered "simulationist" - we can make up flavor text that makes sense, if required. Traditionally, 3E uses the Vancian approach to magic which implies that casters have to prepare spells only a limited number of times per day, and when cast, the preperation is "gone".
4E introduces a vancian-like power system for all classes, even the "mundane" ones like the martial characters.
This can either mean the world of 4E is very strange and hard to "get into" - because we have to assume that a Fighter for some reason can perform a specific combat maneuver only once per day (no matter how hard he tries) - and he might even be aware of it (just like a 3E wizard knows when he's out of spell prepations for a spell). Or we have to at least assume that martial powers are magical (which might be easier, even if not satisfying for everyone).
To avoid this, you can leave the "simulation" stance and thing narratively - the character could in theory do his favored maneuver all the time if the opportunity arises, but the character has no control about the opportunities - but the player of the character does, by using a game resource - the power. (That's the core of it - you can spin a lot of extranous stuff to explain certain aspects, but this should do for now.)
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From my point of view, one major change in 4E is the goal to make the game very usable.
You can pick up the game at any time, any level, and it will be playable with little or no preparation (once you got the basic stuff down).
There is a lot of emergent complexity in the way "in-encounter" tactics might work, but to start, there are not so many things you have to worry about.
You do not have to wade through a a 4 page spell list to figure out what to slot and whether to cast it after breakfast (or for breakfeast -
Heroes Feast), before entering the dungeon, or during combat. The DM doesn't have to skim through rulebooks to figure out what the monsters actually can do with spell-like abilities or spells they might have listed in their stat block.
Of course, this sacrifices some details (like the innocent low level spell available to some monsters that will not get used in combat, but might be used as a twist or hook for a story.) But from my perspective, the usability is more important.
