My vote for what the movies did most successfully in translating the book to film is what Beverly said, but in all aspects. The art absolutely and completely conveyed Tolkein's Middle-Earth, supported and explained and enhanced the story and the characters, and did everything you need a visual medium to do when interpreting a print medium.
The least successful transition was in the characterization of some folks. Whether it was necessary or not, whether it was the right thing to do when making a movie for 21st-century audiences or not, whether it made better characters than Tolkein did or not, the interpretation of Aragorn and Faramir as doubting, conflicted, and uncertain was not Tolkein's intent for those characters. He was creating a mythology, and the Men of Numenor were not three-dimensional. They were stalwart, brave and true, or they were corrupted by an external evil.
Merry and Pippin suffered in the other direction. Again, I think a good decision for the movie in most respects, but you do lose Tolkein's intent of all four hobbits being capable and "in on it" and bridge characters into the mythology.
The movie just missed what would've been the worst possible transition, making Arwen a Warrior Princess, thanks to Liv Tyler's inability to carry it off. FOTR toed that line a little too closely for comfort, but it could've been so much worse.
Normally I would list the absence of The Scouring of the Shire as a big problem also, but I'm not sure it fits the thesis.