A PBBG is not the same thing as a MMOG (massively multi-player online game), generally speaking. There is some overlap, but MMOGs include games that are run as an application on your computer. Application-based games do not have the portability that PBBGs do, because you can only play them on a computer that has the game installed. Think of PBBGs as a subset of MMOGs - those that are browser-based.
A PBBG is also not the same thing as a browser-based game, generally speaking. Browser-based games include arcade, puzzle, and card games which are over as soon as you move your browser to a different web page. These games lack the longevity of PBBGs, where progress is made over days, weeks, or months. PBBGs are a subset of browser-based games - they are persistent browser-based games. Definitions curtasy of www.pbbg.org