Well, I could be snarky and say "usability" -- which, relative to GitHub, it kinda seems like BitBucket pioneered. Frankly, though, I'm not sure whether there's something BitBucket really "invented". There doesn't seem to be much that either can do and the other cannot, aside from BitBucket's ability to make unlimited private projects (of limited size, of course) where GitHub cannot.

Usability seems to be a major difference between not just BitBucket and GitHub, though -- it's also a major difference between Mercurial and Git themselves. Between the two, I find I like Mercurial/BitBucket much more usable than GitHub/Git. The big advantage GitHub/Git has is popularity, basically, and I've never really been one to let a lack of market-dominating popularity hold me back.

Anyway . . . to answer your question more directly:

I don't pay much attention to the fiddly corner cases where they differ in total functionality, so I don't really know who came up with what features first. They've both been evolving pretty quickly, from what I've seen, so I'm not sure anyone else can really say what features one has copied from the other, unless they were working on one of the two and involved in the imitation process.

I don't even see why it matters. It's like open source software: imitation is a good thing, because good features get more widely used, and everybody wins. It doesn't have to be a competition. I just don't think holding up GitHub alone as some kind of messiah really helps anything.