I think there's probably a lot more to it than we know, but I think Axl felt Slash wanted to be THE man in GNR and assume the power.  Most lead singers don't tend to deal well with anyone else being the center of attention.  I think he also feels he tried to work with Slash in certain areas, but Slash was too stubborn to give in.  And ultimately, I don't think Axl likes how Slash has painted him to be the villain in the bands demise amongst other things.  It also probably doesn't help that the vast majority of people tend to side with Slash on these matters.

And obviously, Axl had some major problems with the "lies" told in Slash's book.

Obviously you could flip most of those things around the other way, but I think that's the root of the problem(s).

All in all, I think a lot of it stems from jealousy.  Slash is well liked and respected by most everyone.  It's tough to find someone with a bad word to say about him, outside of these forums at least.  The same cannot be said for Axl.