The obvious advantage of a pseudonym: you stay anonymous if you want to (even if it may become difficult if you contribute to a well known project). This means several things:
- You are not bothered with emails from people who know your name and the fact that you participate to the project,
- If you work in some companies with strict policies, you avoid the problem where all you write belongs to this company¹,
- You can quit the open source project at any moment, and nobody will bother you, never,
- During an interview for a job, you will never have to answer questions like this:
Ten years ago, you contributed to project <Name here>. We've downloaded the code of this period from SVN, and there is a lot of unreadable code, difficult to maintain, with plenty of bugs, no comments at all, etc. Can you explain that, and what changed for the past ten years in your coding style?
At the opposite, it means that if you make a professional-level contribution, writing a high quality code, etc., you're not rewarded personally. You can still say that it's your code during interviews, but it's much more indirect.
¹ I'm not a lawyer, so it may be inexact in your country.