Wired decapitalizes "internet," "web," and "net." Wired has been very influential in how internet-related writing is styled. I wonder how quickly other publications will update their style guides. I blame Wired for the proliferation of "e-mail", though I don't know for sure if that's entirely fair. I prefer "Internet" for its resemblance to a proper noun, vis-a-vis Wikipedia's justification, though perhaps I'm just used to it. I've never used "net" to refer to the internet or any network in any formal writing, and in informal writing I would write "the 'net." On the other hand, I've never capitalized "web," but I don't think I've ever refered to the portion of the net that consists of web pages as "the Web," either, at least not in writing.

The Chicago Manual people answer net style questions.

A single source of formal writing needs a consistent set of stylistic guidelines. The choices major publications make tend to influence a cultural notion of what professional writing looks like, so while style guide declarations are not hard-nosed rules of writing, they're often a pretty big deal. To writers, anyway.