I always capitalize when referring to "the Internet" because it is a proper noun. Of course since 2004, in everyday dialog, I now refer to it as "the internets." Because I only say this and typically do not type it, I never capitalize it (even in my head). And when typing it just now, I couldn't bring myself to capitalize it, although I wonder if the irony would somehow be heightened if i did.
I don't think I ever use "internet" in any context when not referring to "the Internet," but rather, use the term "network."