From Wikipedia, emphasis mine.
"On remand, the District Court ordered Napster to monitor the activities of its network and to block access to infringing material when notified of that material's location. Napster was unable to do this, and so shut down its service in July 2001."
YouTube has been very diligent about removing infringing material when requested to do so via the notification process. Actually, they've been so overzealous that they've had some complaints from the other side: people who have recordings deleted despite their fair use of source material.
The music industry also learned a lot from Napster and early peer to peer challenges. With YouTube, they've really embraced the technology for the most part. Many videos are uploaded by music labels themselves (e.g. vimeo) and have attached commercials. I think if YouTube deletes competition and the labels have a higher quality upload anyway, most people tend to put up with brief advertisements. I also think a lot of bands who make most of their money touring have realized that suppressing bootleg concert videos, which are essentially advertisements for the shows themselves, harms sales of tickets. In short, the music industry has really accepted the role of peer to peer media sharing when it comes short of album piracy. They're still fairly pissed off about things like Bit Torrent.
The studios, on the other hand, are basically where the music industry was ten years ago. It would almost be hilarious if it weren't so tragic that they have to replay the same thought process as the record companies.
Anyways, I think that YouTube can shoot down infringing material from the studios and still have enough to offer users such as music videos, user created content, public domain video, and orphaned works. Napster really didn't have a viable business once they lost all the music. Moreover their more pirating users drifted over to other sharing technologies once it became clear that Napster could not shield them from record labels.