For private projects, GitHub charges by the number of repos and BitBucket charges by the number of users. Initially I was using BitBucket but after having used GitHub I eventually opted to pay more on GitHub. There are two major factors for this. First the UI for GitHub is extremely polished. It's a hands-down winner. BitBucket has always felt aged and clunky in looks and operation.
The second reason is the tie in to open source which makes it very easy to fork open source projects on GH, and then include them in your projects. Again, the network and activity graphs, the commit log, diff/blame/pull-request features etc, just make this whole aspect a lot more fun to use.
I suspect the only reason BitBucket is surviving is because of their differentiated pricing model. If GitHub offered a hybrid pricing (repo or user based), it would mark a sharp decline for BitBucket even though as a consumer I prefer if they were more competitive.
Github offer a free micro account - 5 private repos, unlimited collaborators. Even without a student offer Bitbucket offer unlimited private repos with 5 collaborators for everyone, which is already awesome for students. They'll bump it up to unlimited collaborators for students though, which is a seriously generous deal.
It's not as slick but it's a no brainer for anything that can't be public, which there's a lot of in university.
Also, I use Git to manage large projects written in LaTeX. So far the limit of 5 collaborators haven't been an issue.
Another thing I like about Bitbucket is it far less gamified than GitHub. The latter pretty much has gamification and phony social interaction built into its core, whereas Bitbucket has less bullshit (while still retaining useful things like private messages).
In addition, another more cynical reason to use Bitbucket is as a middle finger to GitHub monoculture. Linus Torvalds once snarked about how Linux being written in C "keeps the idiots out". Using BB is also very effective at keeping the peanut gallery out and having less distraction in general.