Should the words "internet" and "web" be capitalized?

My reasoning is that there is really only one network named Internet - therefore it's the Internet, while "web" is a more generic term, meaning any network (but probably referring to YouTube anyway ;).

I think it helps to know the history of the word Internet.

When computers were first connected together the resulting configuration was called a network.

Later people connected networks together. That was called an internet.

Eventually most of the networks in the world were connected into one large internet that became known simply as the Internet.

In other words, there is a technical distinction between an internet and the Internet.

For what it is worth, my iPad wants me to capitalize Internet.

The fact that Web is short for World Wide Web, Sir Tim's wonderful invention, makes me think that Web should be capitalized too.

It's a matter of style.

The Guardian style guide uses lowercase:

internet
net, web, world wide web

web, webpage, website, world wide web

is there at least some consensus among the more popular style guides/dictionaries on this?

Not really. Wikipedia has a page on the topic, Internet capitalization conventions, with a section on usage:

Examples of media publications and news outlets that capitalize the term include The New York Times, the Associated Press, Time, and The Times of India. In addition, many peer-reviewed journals and professional publications such as Communications of the ACM capitalize "Internet", and this style guideline is also specified by the American Psychological Association in its electronic media spelling guide.

More recently, a significant number of publications have switched to not capitalizing the noun "internet." Among them are The Economist, the Financial Times, The Times, the Guardian, the Observer and the Sydney Morning Herald. As of 2011, most publications using "internet" appear to be located outside of North America, but the gap is closing. Wired News, an American news source, adopted the lower-case spelling in 2004. Around April 2010, CNN shifted its house style to adopt the lowercase spelling.