Doing a backflip off a chandelier to land and stab an orc versus walking up and saying "I attack" are definitely differences in flavor, but ultimately both are pretty much mechanically the same darn thing.
Within the granularity of the system, anyway. In 3.5, there's a difference between moving your speed and attacking, and tumbling half your speed and attack - the latter doesn't provoke. And, 3.5 typically used minis & a play surface, so you might find yourself in a situation where, because of how everyone's positioned and the room is set up, you can't reach a target by moving & attacking without provoking, but you can make a tumble check to get to the same target by passing 'through' an enemy's square - so you make a tumble check, and maybe describe it as a backflip off the chandalier, or maybe not.
If you're running the same scene TotM with no tumbling rules, you can just move & attack, because who know if there's a 'safe path' to the enemy. You can describe it how you like. It doesn't matter.
You can describe the differences however you want but both are 1d20+5 attack for 1d8+3 damage with no extra riders and that is why A LOT of people play spellcasters or Battle Masters.
IMHO, all the real fun to be had in 5e is on the DM's side of the screen.

But, yeah, casters' are your second-best bet.
5e doesn't have detailed rules telling you exactly what you can do and when exactly when you're allowed to do it. This allows for far more complex combat than 3.5/3.75 ever allowed.
Actually 5e does have rules telling you exactly when you're allowed to do things. They're actually pretty tight. You act on your turn. Outside that, you get /one/ reaction. Period. And, if you want to use magic, 5e does have detailed rules telling you exactly what kind of magic you can do and how often you can do it.
Imagine that you're on a raised part of a ship deck playing a pirates campaign. There is a rope loosely "tied" to a pole. You want to do what they do in pirate movies. Grab on to the rope (use an object), swing (use movement), attack, keep swinging (use rest of movement), then let go of rope (no action required). In 3.5/3.75, you'd be told you can't do that because the rules force a specific way of playing.
In 3.x/PF, you'd roll Tumble (DC 15 to swing past the enemy, 25 to swing through his square) to swing up to half your speed in a situation like that. Whether the DM let you attack and keep going is another question, given the nature of swinging from a rope, it'd seem reasonable (yeah, 5e is not the first edition were DMs had to make rulings, just the one where they have to do it every time), of course, if you had Spring Attack, no problem, and you might not even need the Tumble, since you already avoid provoking.
In 5e, you can't necessarily do any of that, the DM has to rule whether you can and whether and how hard any rolls you need to make may be. Your first level fighter might be able to pull it off without making a check (if you're sitting at my table, for instance), while another DM might make it too difficult for your 9th level Rogue with Expertise in Accrobatics to even bother trying. That's actually a strength of 5e, because it lets each DM run with the tone he wants from the game. Not that you can't do that sort of thing in any game, but 5e practically begs you to.
In 5e, you're limited by your imagination.
If you're the DM. Otherwise you're limited by the DM's imagination, and his opinion of you & your imagination...
Some people lack imagination.
Some people resort to insults.