In his brilliant book J. R. R. Tolkien: Author of the Century, Tom Shippey spends a good deal of time trying to account for the depth and intensity of the hatred of Tolkien among the literati. Many of his points are worthy, but I am especially drawn to something he writes near the end of the book, which he is comparing Tolkien to James Joyce — and there are indeed some interesting points of comparison, for instance in the generic forms their great ambitions take and their fascination with language. But of course there are huge differences as well, and Shippey focuses on one of the most important when he notes their radically different attitudes towards the classical tradition. 

Shippey points out that much Modernist writing depends heavily on literary allusion, and especially allusion to the literary inheritance of Greece and Rome. Ulysses is the obvious example here, followed closely by Eliot’s “The Waste Land.” Joyce refers occasionally to Irish myth and legend, and Eliot quotes the Upanishads, but those references are not central; if you really want to get to the heart of those texts, you must know Odysseus and Tiresias. (Shakespeare too.) Even Yeats, for all of his invocations of Irish legend, expects his readers to know about Leda and the swan and to grasp the significance of the death of Agamemnon. The essential works of the classical tradition are the lingua franca for the most ambitious and demanding writers in English-language Modernism. (As they were for Milton, who effectively defines ambition for so many writers that follow him.) 

Tolkien doesn’t care about any of this.

He alludes frequently to works of what he regarded as his own tradition, the ‘Shire tradition’ of native English poetry…. Tolkien's heroes and his major debts came from the native and Northern tradition which Milton never knew and Eliot ignored: Beowulf, Sir Gawain, Sigurd, the Eddic gods — a tradition seen by most modernists as literally barbarous (the possession of people who speak incomprehensible languages). 

In brief, “Tolkien was as educated as [the literati] were, but in a different school.” 

Educated in a different school. And the key point here — Shippey hints at this, but is not quite as explicit as he might have been — is that Tolkien never expects his readers to know any of what he knows. To fully appreciate Ulysses you need to know the Odyssey, but the reader of The Hobbit need not be aware of Snorri Sturluson’s “Tally of the Dwarves” in his Skaldskaparmál:

Nár, Nainn, Nipingr, Dainn,

Bifur, Báfur, Bömbur, Nóri,

Órinn, Onarr, inn, Miöð̠vitnir,

Vigr og Gandálfr, Vindálfr, Þorinn,

Fili, Kili, Fundinn, Váli … 

Indeed, perhaps it is better if we don’t know, at least not until after we’ve read and enjoyed the story. Similarly, it is certainly interesting to note that the exchange between Gandalf & Co. and Háma, the Doorward of Théoden, in The Two Towers is nearly identical to an early scene in Beowulf — but Tolkien doesn’t expect you to know that and your appreciation of the scene isn’t diminished if you don’t. 

The great Modernist writers have a tendency to flatter their learned readers and disdain the others; they are in many respects principially elitist. (As has often been noted, Leopold Bloom is Joyce’s hero but he couldn’t have read Joyce’s book about him.) There’s none of this in Tolkien; the astonishing range of allusions to medieval writing in The Lord of the Rings is certainly meant to provide a kind of felt (not directly perceived) coherence to the reader — Shippey is great on this — but its primary purpose is to satisfy Tolkien’s own imaginative needs. There was, I think, something creatively liberating about having been educated in a school — Germanic and Anglo-Saxon philology — that virtually none of his readers ever attended.