The web is one of several parts of the Internet. Internet is historically older, and includes all these parts:
for one-way fetching of files: Archie, Gopher, FTP
discussion groups, also known as 'newsgroups', although they don't deal with news in the normal sense of the word
the web started to be developed c. 1990 as a way of using the international network of computers as a means of publishing. Since the web's software, mainly Netscape and Explorer, have more and more integrated the other functions, there is a tendency to use the two terms as synonymous.
WHAT DO THE TERMS MEAN?