In many ways a lot of things that are listed as being "iconic" D&D have either been stolen adopted by other games like AC, hit points and ability scores. Sometimes the labels are changed or rearranged slightly but the basic concept is still there. The pseudo-medieval/sword and sorcery mashup feel is also part of it, but that's also not particularly unique to D&D
There's been a lot of great posts on this too but I think talking about what makes D&D I think requires talking about 4E and why many people didn't think it was "real" D&D while 5E returned to it's roots.
I know, I know, this if violating policy. So let me say up front that I don't think the concept of 4E was bad. It was a fine game which unfortunately bogged down at higher levels. However, in all honesty higher level play never had been particularly viable in D&D up to that point anyway. It just changed from "wait until the caster's turn and the fight will be over" to "just wait an hour for this round to be over." It was amplified in 4E because it promised epic level play right out of the box. Queue the "but we played OE D&D to level 50" complaint, I'm just relaying my experience and I think it was a pretty common one.
But despite having the D&D label, to many people it didn't feel like D&D. Eventually I came to believe that as well, and I was a staunch defender for a long time.
So what was different?
Saving Throws
Someone cast a spell at your character? It was almost always the caster rolling with the PCs being passive (barring an interrupt). This seemed to lead to less off-turn engagement, if the enemy wizard was going to cast fireball on you, you didn't have to do anything other than roll a number. Checking to see if you make your save can be a tense off-turn moment.
Sameness of character powers
In 4E everybody had similar structure in how their character's abilities were expressed with at-will, encounter and daily powers. This kind of thing used to be the realm of casters and if you didn't want to deal with it you could play fighters or rogues.
In 5E, different classes and builds share the same chassis but can play quite differently. Want to manage resources? Play a wizard. Just want to hack and slash? Fighter. Something inbetween? Barbarian or the most awesome of all classes, the paladin.
Assumption of a Grid and Fire Cubes/Optional rules
The rules in 4E really did push the idea that it was just a minis tactical combat game when it came to fights. There's nothing wrong with that, but it was very prescriptive in the "correct" way to play. Yes, fire cubes were a result of simplified distance calculations, but they were just one of the things that separated game actions from how things worked in real life.
In 5E, we have a lot more flexibility and there is no one true way. Use a grid, don't use a grid, mix and match, who cares. Want roughly spherical fireballs? There's an option for that. So you can emulate old school (theater of the mind), minis and a grid that started with 3.x. I don't remember grids being discussed in older editions, but we used rough grids for positioning using buttons for monsters and coins for PCs.
Detailed rules and powers
This one is a little tougher to explain, but there never seemed to be as much freedom and creativity in 4E. The rules tried to be very complete and, while I argued against it for a long time, seemed to limit freedom and creativity. It was one of those things I could never 100% put a finger on, but I think it came down to powers that spelled out exactly what you could do. If you wanted to accomplish something that was spelled out in a power the feeling was that you couldn't do it if you didn't have that power.
As much as 5E's style of the importance of DM making rulings to fill in the gaps bothers some people I think it's an important part of the collaborative story telling.
Lack of Detail
I was excited to get my 4E monster manual. But it was just a picture (maybe) and a list of stats with minimal "fluff". There was nothing, as a DM, telling me why a hobgoblin was different from an orc other than what powers they had.
In 5E we get a decent amount of fluff and books like Volo's Guide goes even further to flesh that out.
Conclusion
So that's my list. It's not just one thing. It's not just the mechanical aspects other people have covered, it's also the feel of the game. Enough structure to have a consistent experience, enough freedom to encourage imagination.
P.S. I'm not going to quibble about details of 4E. These were just my impressions and experiences.