No, the Web is one way to use the Internet, but the Internet is more general than the Web.
The Web is a set of web servers (programs like Apache and IIS). Your web browser (programs like Firefox, Opera, and Explorer) connect to these web servers using the http protocol. A protocol is a set of rules about how to communicate information. The http protocol is a relatively simple protocol to exchange information, mostly in one direction, from the server to the client. A similar protocol is https, which is like http except it encrypts the data being sent. When you buy something over the Web hopefully your web browser and the web server are using https instead of http.
There are many other protocols that are commonly used, other than http and https. In the same way that your PC is more than just solitaire, the Internet is more than the Web. One of the most important protocols is the simple mail transfer protocol (smtp) which is used to transmit mail from one place to another. Even e-mail sent from one web e-mail account (like yahoo, google, or hotmail) to another uses the smtp protocol. Most online games have their own dedicated protocol. The reason there are so many different protocols is that they are each designed with different goals. Using http to play World of Warcraft would make it much slower. Likewise, bittorrent would not be a good choice to write an instant messenger program. So, although the terms the Web and the Internet are often used interchangeably, the Internet is the infrastructure that supports many different forms of communication, and the Web is just one of these forms.
Originally published in Standard Output, Vol. 2, No. 1, February 2007.