Games these days (and I don't mean to speak broadly because not all of them are like this) ask you to follow a pre-ordained character with already chosen characteristics through a two-dimensional storyline that sometimes only has one ending. It's all very linear. You aren't given the opportunity to change the path of your protagonists by making different actions that alter the end result, you merely sit through whatever it is the game has to offer you when all is said and done as though you are a helpless spectator watching a game of football.
Heavy Rain wasn't like that. Take for example the case of Norman Jaydon, one of the four protagonists in the game. I think it is safe to avoid putting up spoiler alerts given the age of the game, but I will anyway. *Spoiler Alert*
There, still here? Good. Norman Jayden is an FBI profiler first and a drug addict second, hooked on the seemingly powerful and absolutely fictitious triptocaine. Instead of sitting through his decline and watching him hit rock bottom, the gamer is actually provided with the option to alter his destiny by resisting his urge to inhale the triptocaine through a series of prompts (and a bit of button mashing) on the D-pad.
There's absolutely more control in games like Heavy Rain and more games should follow suit in that department.