For me the biggest problem is that it doesn't clear upon reload. Having to keep track of which messages/errors came from the previous page is really annoying.
Edit: Also, viewing objects is inferior compared to Chrome's tools.
You might want the ability to clear console on page reload as an option. It seems useful and people should be able to enable/disable it (or was it added to new nightly).
I can because I just found that very useful. Think of code that has an error in an window.onbeforeunload handler. By the time the console displays the error the page has switched. Without being able to view the errors from the previous page, this would have been impossible to catch.
Edit: Also, viewing objects is inferior compared to Chrome's tools.