Basically Peter Parker/Spectacular Spider-Man and Amazing Spider-Man are two seperate series. They each have their own stories that don't effect each other. Major things, like is Mary Jane broke both her arms, would be mentioned, but that's about it.
The one I read is Amazing, as it's written by J. Michael Stracynzki and he does an amazing job. The artists on that series (John Romita Jr and more recently Mike Deodato) are amazing and I highly reccommend it. You really won't need to go out of the way to pick any back issues up to get acclomated with what's going on, maybe an issue or so before just to have the complete story arc.