Here is a pretty good video comparing the two and explaining what TWD did right that Heavy Rain failed at: https://www.youtube.com/watch?v=r6gC2MHXIH0
[spoilers for both games of course]! Personally I haven't played Heayv Rain but I have watched a whole playthrough of it and I think the game has a very interesting story it just fails at making you care a lot about the people in it. Needless to say, the game was still good. TWD however is much much better.