What happened to Budd?

Well, first of all the following is entirely speculation by me, since we don't get too much insight into his backstory in the movie. It might only be a first start, as I might have to rewatch it to further substantiate any of this with some specific dialogue or other hints.

That being said, it seems to me as if Budd pretty much wanted to distance himself from Bill's whole business and that previous life as a world class assassin (if he ever was one). He seems a bit disillusioned to me. For example when he says to Elle something like (forgive me for not recounting it word-accurate):

This woman deserves her revenge and we deserve to die for what we did to her.

In the same way, if I remember correctly, he said something like that he sold his Hanzo sword, an item of absolutely unestimable value to any of the others in Bill's crew. So he obviously wasn't engaged in the matter with his heart and soul anymore (if he ever was).

It is unclear, though, if it actually was their assassination of The Bride that caused this change of mind in Budd or if he never really was so much of a real member of the gang. To me it rather seemed that he never was so much devoted to the whole business let alone such an established member as the others, but was primarily part of the gang because he is Bill's brother (neither is he a woman nor a Kung Fu magician either, if I remember correctly).

First of all, Budd's Hattori Hanzo sword was from Bill, so there's no indication it came from Hanzo himself.

Second, we never learn why he & Bill are estranged.

This is pure speculation, but I think Budd had a gambling problem, so when the group disbanded (which I think happened after they killed the bride & co because it sent Bill into a depression), without a leader they all kind of went their own way, with Bill as sporadic leader/snake charmer/wrangler on assassin jobs... Budd seems like much more of a fuck-up than Bill, so without Bill's leadership, he probably spiraled himself, but with gambling, drugs, women & booze... With no money, and unwilling to sell his Hanzo sword (the gift from Bill, not black mamba's sword), probably for sentimental reasons.

Tarantino is great at character development, so I bet he has answers for all this stuff. I wish he'd make a Kill Bill version of Animatrix or Clone Wars (animated or live action), to fill in some of the questions.

Side-note: the reason I think he kept his sentimental sword: because most drunks/gamblers are emotionally stunted & sensitive.