The internet is the scaffolding/structure, the Web is what people are doing in a browser (i.e., over HTTP) in it.
Then there's also the stuff people do on the internet without a browser/HTTP. Nobody opens an IMAP/SSH/BitTorrent/IRC client or whatever and thinks of that as surfing the Web, because those aren't browsers nor are they primarily speaking HTTP.
I'm not sure, but the grand-parent might be drawing from Hakim Bey's distinction between Net and Web. This is from TAZ, The Temporary Autonomous Zone (1991):
We’ve spoken of the Net, which can be defined as the totality of all information and communication transfer. Some of these transfers are privileged and limited to various elites, which gives the Net a hierarchic aspect. Other transactions are open to all — so the Net has a horizontal or non-hierarchic aspect as well. Military and Intelligence data are restricted, as are banking and currency information and the like. But for the most part the telephone, the postal system, public data banks, etc. are accessible to everyone and anyone. Thus within the Net there has begun to emerge a shadowy sort of counter-Net, which we will call the Web (as if the Net were a fishing-net and the Web were spider-webs woven through the interstices and broken sections of the Net). Generally we’ll use the term Web to refer to the alternate horizontal open structure of info-exchange, the non-hierarchic network, and reserve the term counter-Net to indicate clandestine illegal and rebellious use of the Web, including actual data-piracy and other forms of leeching off the Net itself. Net, Web, and counter-Net are all parts of the same whole pattern-complex — they blur into each other at innumerable points. The terms are not meant to define areas but to suggest tendencies.
One system built on the Internet is the World Wide Web, which is just webpages served with the http/https protocol.
Other protocols that route over the Internet include email, ssh, Tor, torrents, apps, etc.