The "web" part runs atop the "Internet" part, where web is at the application layer (those lines can blur on that), and internet is the lower levels of the OCI Model. See image.

The World Wide Web is more or less the Internet as most users see it - HTML documents located via URLs shared over the HTTP protocol. But that's just one specific protocol used for sharing a specific kind of content (hypertext). It turns out you can do a whole lot with that, hence the ubiquity of the Web.

But the Internet as a whole is broader than that. There are other protocols, other content to share, other ways to locate data. For a down-to-earth familiar example, just consider any online multiplayer game. You're using the Internet to communicate to the game's servers to play the game, but that Internet traffic is certainly not part of the Web.

When you're playing an MMORPG you're not using the web, but you're using the Internet. The Internet is like the postal service relaying stuff, but the stuff can be of different kinds.

The Web is websites (HTML pages served over HTTP, which link to each other via hyperlinks). The Internet is websites plus email, VoIP, IRC, Usenet, bittorrent, game servers, FTP, SSH, and everything else that isn't a website.

Web is just the website part. Like web pages you see in the web browser. Tim Berners Lee invented this part.

Internet encompasses more things. For example, sending texts in some mobile app. You’re not necessarily seeing a web page. You’re just sending data from your device to some other device in the world.

Or to be more technical: web is HTTP. Internet is that and everything else (like FTP, SMTP, SSH, etc).

Or more simply: The internet is just the network. The web is one of the handful of "apps" that run directly using that network. It just also happens to be able to run apps of its own these days and is a pretty "killer" one at that :)

(Edit: please disregard the wrong link that summoned the bot below. It's been fixed. And I do not have a Balatro problem...)

Great way my professor explained to me with:

Internet is a network of computers, to send data to each other.
Web is a network of documents, the revolutionary concept about it was the fact it can be on your computer or others computers and still be connected via URLs.

When asked to describe what distinguished him from his challenger for the Democratic presidential nomination, Senator Bill Bradley of New Jersey, Gore replied (in part): "During my service in the United States Congress, I took the initiative in creating the Internet. I took the initiative in moving forward a whole range of initiatives that have proven to be important to our country's economic growth and environmental protection, improvements in our educational system."

-Snopes

His choice of words was poor and makes it sound like alone “took the initiative in creating the internet.” Whatever he really meant, his wording makes it sound like he’s claiming a bit more credit for his role than it merits.