Many folks consider that they both are same but are they really same?
The Internet is a global network of networks while the Web, also referred formally as World Wide Web (www) is collection of information which is accessed via the Internet.
Another way to look at this difference is; the Internet is infrastructure while the Web is service on top of that infrastructure. Alternatively, the Internet can be viewed as a big book-store while the Web can be viewed as collection of books on that store. At a high level, we can even think of the Internet as hardware and the Web as software!