You’re right to a point. The Internet is much older than the Web. Content on the Internet was largely FTP sites, Gopher sites, and Usenet long before the Web existed. We also had MUDs, MUSHes, MOOs, talkers, chat rooms, BBSes, and more over Telnet well before the web was widespread.

    Telnet    1969, standardized 1983
    FTP       1971
    UUCP      1979
    SMTP      1981
    RSH       1983
    TCP/IP v4 1984
    NNTP      1986 (the successor to UUCP for newsgroups)
    IRC       1988
    Archie    1990 (search engine for ftp sites)
    Gopher    1991
    WWW       1989 first published 1991
    Veronica  1992 (search engine for Gopher sites)
    SSH       1995

And in parallel we had FIDONet. Mustn't forget that magical place and source of many a late night and large phone bill.