Email (not webmail, but IMAP and POP etc) uses the internet but is not part of the web. There are other examples but I think that's the easiest to see.
"The web" isn't exactly a well-defined concept. There's a school of thought in which anything with a URI is part of the web.
Emails don't have URIs. My email boxes don't have URIs. My main problem is that there's no way to link emails together, so they're not really part of any web.
Does mailto:person@example.com not count as a URI to your mailbox?