Epic Rap Battles of History a while ago staged this battle between JRRT and GRRM.
How did the fantasy genre develop from Tolkien's time? Is GRRM really an innovator in the genre? Are there other authors we should consider as developing the genre?
I think this is an interesting question, especially considering how much attention Martin has gotten in recent years. Maybe even more so now that the HBO series is over and people are looking back at Martin again to "salvage" the story. I can't promise to be able to say much about the entire period between The Lord of the Rings and the A Song of Ice and Fire novels, a period of some 42 years with hundreds of authors writing everything from drug store novels to critically acclaimed major works during that time. But I think I can claim to know a few things about the period in the development of "soft science fiction" and fantasy immediately after The Lord of the Rings, so basically the 60s and the early 70s, which is incidentally one of the most fruitful periods of so-called speculative fiction.
In my experience a lot of people are a big surprised to hear that The Lord of the Rings was only published in the mid-50s--The Fellowship of the Ring is 1954, and the trilogy was ended a year later. By comparison C.S. Lewis, probably the next-most famous member of the Oxford writing group (basically a club) that called themselves the Inklings, of which Tolkien was a member, was already well established as a fantasy writer. Lewis had published several works pretty early on, but his big magna opera in the fantasy field were the so-called Space Trilogy (really a fantasy trilogy, the first novel of which, Out of the Silent Planet, was published in 1938) and of course Narnia (The Lion, the Witch and the Wardrobe is 1950). That generation of fantasy writers was already well in place when Tolkien published The Lord of the Rings. Meanwhile, the distinction between science fiction and fantasy was still surprisingly recent: Robert Heinlein, to my knowledge, was the first writer of speculative fiction (a term he claimed to have coined in the 40s) who formally made a distinction between fantasy and science fiction, although in the 30s and 40s there was a lot of redefining of terms and introduction of new jargon to describe what was going on in these imaginative works of fiction. Nonetheless, the influence of The Lord of the Rings I think is immediately obvious to anyone who's ever picked up a fantasy novel written since then. It looms over the genre, distinguishing work written before it, like Robert Howard or C.S. Lewis, from stuff written after it. To that extent, Martin is very much in the Tolkien camp, and the influence of Tolkien I think is obvious enough that I leave it to the critics to point it out.
Tolkien's work defined what is now commonly called "high fantasy," or "epic fantasy," a term that doesn't really mean very much. High fantasy differs from so-called "low fantasy" and the early "sword and sorcery" work of people like Howard in a number of ways, although I think what you and I would pick out most obviously, as classicists, is the obvious influence of epic literature. A hero or group of heroes undertakes some great quest, potentially not of any immediately obvious significance but at least eventually encompassing the fate of the entire world, and broadly follow Joseph Campbell's paradigm for the Heroic Journey as used in epic. There are definitely bits of this in earlier literature--it pops up in C.S. Lewis, although he's more interested in allegory, which Tolkien insisted had nothing to do with The Lord of the Rings--but in the late 50s and early 60s this became the way to write fantasy. Just as science fiction by this time had really developed into its own unique form, fantasy began doing so under the influence of Tolkien.
The late 50s and early 60s also saw, on the other side, the beginning of the so-called New Wave in science fiction. The so-called Golden Age of science fiction, between the late 30s and early 50s, saw the rise of "hard science fiction," the crunchy stuff written by people like Arthur C. Clarke and Isaac Asimov. By the 50s science fiction had begun to transition from a genre confined to magazines to stuff that could actually be published in its own books. But interestingly, just as fantasy was beginning to be the sort of literature that was taken seriously, following Tolkien, science fiction writers were starting in the late 50s and early 60s to turn away from the hard science fiction that had come to define science fiction writing, in a movement sometimes called the science fiction New Wave. The emergence of "soft science fiction" in the early and middle 60s began to overlap with the direction in which fantasy was heading, and they really start to become inseparable for a while, with many authors of this period identifying as one or the other but writing work that can only reasonably be considered what they said they didn't do, and lots of mixing. This era of science fiction was interested in psychology, and in science fiction as a vehicle for many of the same epic themes that fantasy after Tolkien was starting to look at--both benefited from the fact that large publishing houses were starting to consider publishing science fiction and fantasy novels. The mid-60s saw a boom of work that walked the line between science fiction and fantasy, even as many authors insisted they were doing one and not the other.
1965 was of course the date of the publication of Dune, in which Frank Herbert used the politics and structures of quasi-medieval fantasy in an extraterrestrial context. Dune was science fiction technically, but at the time and later people considered it closer to epic fantasy: Brian Herbert, Frank's son, describes Dune not as science fiction but as a modern mythology, a "conglomeration of familiar myths." In this way, Dune resembled Tolkien a lot more than Asimov's Foundation, which, with its grand political interests on a galactic scale, is often compared with the politics of Dune. Similarly, Dune plugged into a trend that was emerging in science fiction and fantasy in the early 60s of constructing complex fictional worlds with their own sets of rules and norms. You'd seen a bit of that earlier (Howard had an elaborate world for Conan to play in), but in Dune Herbert worked out the ecology of the world, its galactic politics, and complex social interactions going even back into the fictional distant past. The same year saw the original (abridged, under the title "...And Call Me Conrad") publication of This Immortal by Roger Zelazny, who would go on to write the seminal Lord of Light two years later. Running with the mythological theme, Zelazny constructed a picture of Earth in the far future through the eyes of an immortal being (who may or not be the god Pan) who sets out on a heroic task to drive out a race of invasive aliens without their even realizing that he's doing it. In Lord of Light Zelazny would take this future mythology further, creating a surrealistic extraterrestrial world in which the marvels of science, long forgotten in the past except by a select few, have made the Hindu gods a reality. In 1966, the year that the full version of This Immortal was published, Ursula Le Guin published her first novel, Rocannon's World, with Planet of Exile and City of Illusions coming right on its heels in 66 and 67 respectively. These novels were part of the so-called (later) Hainish Cycle, which had already been introduced in stories earlier but never really worked out fully. Le Guin constructed a galaxy with an ancient, in many places long-forgotten, history, and used it to explore anthropological and psychological questions (her father was an anthropologist and her mother was trained as a psychologist) by presenting a world that was simultaneously uniquely well thought out but also never really explained except incidentally. It differs from works like Dune in this respect, in that anything that's familiar in the world Le Guin quickly contradicts, by for example mentioning quite incidentally that the barbarian inhabitants of Rocannon's World, which we expect until like a third of the way through to be quasi-Norsemen based on their language and society, are in fact dark-skinned. Le Guin's A Wizard of Earthsea was published, almost by accident, in 1968, and unlike the Hainish novels was devoid of any visible traces of "true" science fiction: there are no traces of science in the world of Earthsea, only magic, try as hard as we might to find them and distinguish the one from each other. Still, though Le Guin was largely not interested in titanic epic plots (although Earthsea especially is epic) she nonetheless used many of the same themes that Tolkien had introduced. Le Guin did not dispense with stories about epic journeys, comings-of-age, and so forth, but added entirely new perspectives on them in worlds that were as well thought out as Herbert's but were basically incidental to what she was really interested in.
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Not to detract from the great answer by u/XenophonTheAthenian, but you might also be interested in what u/AncientHistory has to say in What was fantasy literature like before Tolkien, and how does it compare to Tolkien's works?. It's important to remember that Tolkien wasn't working in a vacuum, and that George R. R. Martin was highly influenced by fantasy writers like Robert E. Howard, Fritz Leiber, and Poul Anderson.