First and foremost, the character of Gollum. The films portray him as a simplistic ghoul with laughably cartoonish MPD and, by the end of The Two Towers, no reedeming qualities whatsoever. In the novel? Well, firstly his dialogue is a lot less heavy handed. Secondly, the multiple personality disorder, whilst still there, is a lot more subtle and believable. Indeed, Gollum's psychology is, I think, one of the most interesting parts of the trilogy. Tolkien, whether he knew it or not, succesfully managed to capture and portray the character of a wretched junkie, albiet a junkie whose drug is a magic ring. Considering that the novels were written in the 1950s and marketed to children, this is a bold bit of writing on the part of Professor T. Finally, even in Return Of The King, Gollum is constantly fighting with himself over Frodo and the Ring, unable to give it up, yet also unable to murder Frodo and Sam outright. I'm sure all fans of LOTR will remember the scene in ROTK (omitted in the movies!) where Gollum almost repents, and Prof. T beautifully describes how far he has fallen, and how beneath his ghoulish appearance he is simply an old, weary hobbit.
Secondly, the scouring of the Shire. Again omitted from the films, yet to me this is the most important part of the story. The hobbits have been away on their mad adventures, far away from home, and managed to defeat evil. They come home laden with praise, and expect to return to their normal comfy lives, and what happens? They find that, whilst they were away, the Shire was taken over by Saruman, and the hobbits reduced to near slavery. From a storytelling point of view, this is the entire reason Frodo and company went away in the first place: to develop the skills needed so that they can liberate their homeland without the need for Gandalf and Aragorn. Looking at Merry, Pippin, Sam and Frodo, and comparing them at the start of The Fellowship and the end of ROTK, it's clear that they've all been to hell and back, but it's also clear that they're no longer four well-to-do gentlehobbits. Each of them is a hero, and the way they each rouse their fellow countrymen to arms is a beautiful thing to behold.
Another thing people seem to moan about is the fact that the books are so long, and it takes a while for the main quest to get going. Again, I don't see why this is a problem. It isn't until the second book of The Fellowship that the quest proper starts, but in the build up to that, Tolkien has managed to lay out the world and the context in which it takes place. We learn about the Shire, the western realms of Middle Earth, the history of Sauron and the Ring, and the terrifying nature of the Nazgul. Then the quest proper starts, and we now know why it mustn't fail. If Frodo had simply headed straight to Mordor from his own front door, we wouldn't have a clear idea of who Sauron is, who the Nazgul are, or why it's really all that important. As well, Tolkien builds up the quest in the best possible way: in stages of gradually increasing magnitude. Frodo's first task is to simply make it to a pub in a not-too-distant village. Once there, the stakes are upped, and he has to slog it cross-country to Rivendell. Once he's there, the story has built enough momentum for the true quest to begin. There's been enough adventure already so that the story doesn't hot the ground running, but the scale of the quest is such that it eclipses what has come before. Again, if Frodo had simply made straight for Mordor, the pacing of the series would have been ruined.
I'm sorry if this has turned into an epic rant, but I'm getting fed up of people trying to make Tolkien out as a lousy writer. If he doesn't do it for you, that's fine, but the man wasn't just a gifted world-builder, he was a skilled author too.