“The World Wide Web” is the name of a unique entity and is thus written with initial capital letters. “The Web” is the short form of the name. In MLA publications, the capitalization is retained in the short form for the same reason that, for example, baseball fans refer to the San Francisco Giants as “the Giants,” not “the giants.” Lowercasing these short forms would imply that the Giants were literally giants and the Web was literally a web. Rather, the names are metaphors. (Tim Berners-Lee, the inventor of the Web, originally considered calling it the Mesh and the Mine of Information—names based on other metaphors .)
The familiarity of the name has no bearing on this question. No matter how often sportswriters refer to “the Giants,” they’re never tempted to shift to “the giants,” because the metaphoric basis of the term is inescapable.
Berners-Lee, Tim, with Mark Fischetti. Weaving the Web: The Original Design and Ultimate Destiny of the World Wide Web by Its Inventor. HarperCollins Publishers, 2000.