No, they are completely different types of games. Alan Wake is an adventure shooter with some puzzle elements while Heavy Rain is more like an homage to the classic Lucas Arts adventures with some action sequences added to it.
Both could be classified as thrillers though and both have very good storylines.
Not really. Alan Wake, as far as gameplay goes, is probably closer to something like Resident Evil, Silent Hill, or Dead Space. Also, the story is far more supernatural in tone in Alan Wake than in Heavy Rain.