The word for the (World Wide) Web is 'Gwe' or 'Y We' (soft mutation after definite article') which just means 'the web'.
As a former IT teacher in a Welsh medium school I used be very pleased with the children talking about 'mynd ar y We' (going on the Web) rather than the incorrect 'go on the Internet'.
This usage is not correct. The World Wide Web is a subset of services using the Internet. Personally I was using the internet (e.g. email, telnet, ftp and even gopher - a precursor of http) since the mid eighties, several years before Berners-Lee invented the Web. There are a great many other services using the internet infrastructure that you would be wrong to describe as the Web (taking a few at random - ssh, ntp, wmts, and many many more). Use of The Web / Y We when the wider Internet / Rhyngrwyd is meant should be discouraged.