James Horner, composer on the first Amazing Spider-Man movie, in which Webb directed, offers why he didn't return for the sequel (Hans Zimmer did the score), and also says Amazing Spider-Man 2 was "dreadful" (via James Horner Film Music).
Horner says because Marc Webb was considered inexperienced, the producers on Amazing Spider-Man (I'm guessing Avi Arad and Matthew Tolmach) had their own opinions and - get this - didn't even want Webb's input.
He was very inexperienced and he and I had a very good relationship and the producers had their own opinion. And they didn't want his input.
Horner then went on to state that all Sony wanted was a bunch of action in the movie.
And then Sony had their own, they just wanted action.
Horner continues with mention there was potential to write good music for Peter Parker and Gwen Stacy, but when he received details for Amazing Spider-Man 2, he decided against it - again - because it was so bad.
To me, the whole thing about doing the movie was I liked the director and there was a chance to write something for the two lead characters and then she dies in the next movie. But the next movie ended up being so terrible, I didn't want to do it. It was just dreadful.
Instead of doing a third Amazing Spider-Man film, Sony has opted to do a Sinister Six movie, which is supposed to be released November 11, 2016. Plans are also underway for a Venom movie and a potential Spider-Man spinoff starring a female character.