The Internet had been around for years before the world wide web and is the set of technologies beneath the web which enable the web to exist. The web cannot function without the internet, but the internet can function without the web.
The Internet technically began to exist the way we know it in 1983 when its predecessor, ARPANET began using TCP/IP. Transmission Control Protocol/Internet Protocol is the framework for the internet system of network communication still used today.
Other programs that use the Internet, but have nothing to do with the web are email, Internet Relay Chat, internet messaging programs, newsgroups, BitTorrent, telnet, FTP, etc.
The web was invented by an Englishman, Tim Berners-Lee in 1990 after years of effort. It did not come into wide-spread use for a few years after that. The World Wide Web is made up of servers, which serve the pages and clients, like Firefox, Chrome, and IE which display the pages. Hey man, I'm on the web tweet me.