If I was talking about the WWW project (or TheProject.html, as the URL used to be), I would say "the World-Wide Web". Using that name to refer to the ubiquitous database that grew from that project is like referring to water as, I don't know, oxygen dihydride (this may not be accurate, since I'm not a chemist).

As for the Internet/internet thing, I write it with a capital because a) it's a proper noun and b) it's not any of the other internets out there. I am inconsistent about Web/web, however.