Tough one, Burnout Paradise has quite a few major flaws, the main one being the open world which for me just doesn't work as well as having dedicated tracks to learn. Heavy Rain was fine, but I don't see it as the pinnacle of storytelling in videogames, and since it is essentially a game of QTE's the connection you have with the story is probably more important in Heavy Rain than almost any other game.
I think I slightly prefer Burnout, but I'm going to vote for Heavy Rain as I feel that is the more highly regarded and therefore deserving of the two to go through.