The largest complaints I've heard from Tolkien fans concerns Martin's world building and overall narrative arc. As a fan, I'd argue his world building is solid, especially in the last couple books. But Tolkien created entire languages for races, and detailed histories that go back thousands of years. Sure, Martin has some of that in terms of family histories, and often says he likes telling a story backwards and forwards so readers know how the world began. Yet it's safe to say the sheer detail doesn't match Tolkien's work. Martin would definitely agree with that assessment as well.
In terms of narrative arc some people prefer the simple, consistent story of LOTR over the more scattered one Martin has created.
Finally there's the argument over prose and writing in general. Tolkien was a linguist and master of the English language, many of his fans would argue his works are simply better written than Martin's.
(personally I like ASOIAF more than LOTR)
