First and foremost, if you haven't seen it already, take a read through KB#1056: A Comparison Between BuildMaster vs. Jenkins; it gives a good idea of the differences between release automation and continuous integration.
Regarding your question ...
What I don't understand is what is the advantage of having BuildMaster take over the pipeline with Jenkins doing the build instead of having Jenkins do it?
... it's really like asking, "what are the advantage of having a people-driven release process instead of having continuous builds automatically tested and then automatically deployed from source to production."
The answer really depends on the organization. To many, the idea of continuous deployment is terrifying; for others, it's satisfying.
What parts in the Blue Ocean screen shot would still be done in Jenkins and what would be done in BuildMaster?
In this case, you would use Jenkins to create a release-suitable build; so your pipeline would look more like Build > Run Unit Test > Run Selenium Test > Export to ProGet > Pull into BuildMaster > Deploy to Integration.
The "Deploy to Integration" would be triggering BuildMaster. Of course, Jenkins can deploy (or really, "do anything" since it runs scripts you write) -- but BuildMaster's deployment capabilities will almost always be superior.
From there, BuildMaster would handle your release process, and actually get it to the proper environments securely, etc.