browserid.org ("the BrowserID popup") is just a way to bootstrap the system. The idea is that browsers and email providers will support this protocol and browserid.org will be totally unneccessary.

The major difference is it's totally irrelevant to the site (relying party) what provider you're using. The site doesn't need a login page with a facebook button, a twitter button, a livejournal button, etc., it just needs a "sign in" button.