The web is the network of hypermedia documents, the internet is the network of computer networks. A "web page" is a node in the graph of the web. A "computer" is a node in the graph of the internet. For practical purposes, the "internet" is the things that link together using http://en.wikipedia.org/wiki/Internet_Protocol and the "web" is things that link together using URIs.

The web is content, the net is machinery.

When you are on Xbox and playing against someone else, you are playing over the internet but you are not playing over the web. Similarly, you can download web pages but you cannot download the internet.