Short Answer: Because that's how the story goes; Don Corleone is shot 5 times and survives, but Barzini is shot only a couple and dies (I forget how many times he's shot in the novel, if it even mentions it)

Long Answer: Maybe the two hired thugs were nervous, and their aim was a little off? They couldn't really stick around and make sure Don Corleone was dead; their mission was probably to ambush him, shoot him a few times and get the hell out of Dodge.

Neri, on the other hand, was a pig (i.e. a cop) and probably a much better shot, not to mention the fact he shot slowly and probably aimed for certain areas, although I have to admit that scene is corny because he takes his time to line up the shots, instead of just blasting.