Why do I get a link to the Mod CP next to the User CP link?

It's just a part of the theme that doesn't hide the Mod CP. You can't use it unless you are in fact a Mod.

Yeah, I found that particularly disappointing when it first appeared. I was like 'holy crap, I got promoted!' Then I clicked it and it didn't do crap.

Closing this because the question's been asked quite a few times. Everyone has the Mod CP visible, but no one can use it unless they are a mod.