There are a few parts to your question, and I'm not sure which one is most important. If the question is about the value of version control, then you should see the answers to this question (which also has a link to a question about which one to use).
If you're asking about putting it on a shared repository, then here are some reasons:
- collaboration with others. If multiple authors are writing a paper, then it's very easy for different authors to edit different portions at the same time without too much worry about conflicts.
- portability: I work from different machines, and it's convenient to have a single location to check in/check out from so I don't have to worry about edits on my home machine versus my office machine versus a laptop and so on.
If you're asking about making it public, so anyone can read/edit the paper, then I don't see much of an advantage in that. When the paper is done, it might be handy to be able to download the source of course.