I prefer bitbucket but also use github.
Bitbucket - prefer the web UI, like the approval feature for PRs, free private repos. For work purposes, we use Bitbucket because it integrates with tools we already use; plus a few team members like Sourcetree.
Github - visibility for public projects (people conflate github=git=open source so that's where they look), auth (eg. this site allows auth by github but not bitbucket), tools that only work with github, or just to contribute to projects hosted on github.
How I suggest people choose: evaluate integration with the specific tools you use; compare the features you use heavily (eg. pull requests); and think how seriously you expect third party contributions from the open source community (who tend to stick to github).