It is important to define our terms, most especially for words that everyone thinks they understand. The terms web and Internet are not synonyms: they refer to completely different things.
The internet is “the network of networks”. It is the physical hardware: the servers, cabling, routers, modems, switches and clients (computers) that combine to form the massive network we call the internet. To use an analogy, the internet is plumbing: it is the physical infrastructure that allows information to flow.
Furthering the analogy, that information – the data – is the fluid that is pumped through the plumbing. In reality, plumbing line could be used to transfer any kind of fluid or gas: in your house you are likely to have plumbing lines for hot water, cold water, and natural gas. Similarly, the internet can “pump” any kind of digital information. In our analogy, the data that makes up the “World Wide Web” would be hot, fluoridated water: it is simply one kind of information that can be transferred through the network. While the web remains the medium by which most people experience data on the Internet, it is by no means the only one. eMail, for example, can be transferred without using the web. (Yes, you need the web for gMail and Hotmail, but Outlook doesn't need the web in order to send eMail). Similarly, you can play a networked FPS game via the Internet without using the web. It is entirely possible to have a working internet connection but not be able to access the web. The reverse can never be true.
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