You guys claiming that the writing IS the difference are really doing a huge disservice to point-and-click adventure games. Of course, the writing in LucasArts games was infinitely better than Quantic Dream's, but that's besides the point. The gameplay is also fundamentally different.
And yes, puzzle-solving is gameplay. You can argue that watching a cutscene and pressing random buttons as they appear is also gameplay (and sure, you interact with the game to a certain extent), that requires almost no thought at all and, to most people, sequences like that are pretty uninteresting.
Now, let me give you a very concrete example. Let's say you get to a hotel and you need to make your way to a certain room to talk to someone there, but the clerk in the lobby won't let you pass.
In an old-school adventure game, you'd probably be able to try talking to the guy, with different dialog options to attempt to convince, cajole and/or threaten him in various ways. If that leads nowhere, you could perhaps try to use physical force, which your character may or may not be willing to do. If not, you can look around the environment for a way to succeed, maybe find a way to distract the guy somehow and go past him while he's not looking. Or ring the desk bell until he gets so annoyed he just lets you go so that you'll stop. Then you also of course have your inventory of items you're carrying. Maybe you have some sort of symbol of authority to just make him do as you please, or you might have... ah, yes, your wallet, I bet this guy would just take a bribe. In short, you have a multitude of options that you have to think of and try, and when you eventually succeed, you get a sense of accomplishment for the obstacle overcome, and you feel smart (if the puzzle is well-designed). Also, even if you couldn't solve that puzzle at that moment, many games from the classic era would offer different puzzles in multiple areas at the same time, so you always had something else to work on.
On the other hand, in Heavy Rain - and yes, literally in Heavy Rain, because this is a scene from that game - you get a prompt telling you to move an analog stick a quarter-turn, and when you do, your character automatically takes out his wallet to bribe the guy and you move on. No player agency whatsoever. Sure, you press a button, but there was no other choice but to do that. Is that in any way satisfying?
Now, if you feel the two approaches are similar in their quality of gameplay, interactivity and player agency, or even in their core philosophy, uh... I don't know what to tell you. Sure, not every scene in David Cage games is that simplistic but... then again, using your wallet would also be the most basic of puzzles in any LucasArts game.