I've just finished a re-read of The Lord of the Rings, and although I love these books, I think there are a couple of places where the Peter Jackson movies do a better job than the books.
In the books, the opening is a bit tonally weird, until we meet Strider and leave Bree. Probably the weirdest bit is the Talking Fox, but the diversion through Crickhollow and the Old Forest, as well as Tom Bombadil, also stand out as tonally different from the rest of the story. They're a bit more of a romp, with less of a feeling of threat than later encounters.This is probably because Tolkien wasn't quite sure if he was writing a pure sequel to The Hobbit or something different until he got to Rivendell and Elrond enters the story.
Bombadil is not forgotten forever after the Barrow-Downs. He gets mentioned at the council of Elrond, and in the epilogue Gandalf mentions he needs to have a long talk to Bombadil while the Hobbits return to the Shire. But he stands out as not particularly relevant for the rest of the story.
These two chapters take place after the Battle of Pelennor Fields, while the leaders of the West debate what to do and end up marching on the Black Gate as a means to distract Sauron from Frodo and Sam. In the movie, Aragorn plays the leading role in this debate, he leads the army to Morannon, and when they arrive, he inspires the Men of the West to fight the hopeless battle. Aragorn's role here as the bringer of hope takes over from Gandalf's role throughout the rest of the book after Gandalf's seeming despair during the battle itself, when he says There never was much hope; just a fool's hope
. Gandalf has to leave Middle-Earth in the epilogue, and the Last Debate lets Gandalf pass the torch of hope to Aragorn (and therefore to Man) before he sails away for good.
In the book, Gandalf still calls the shots in the Last Debate, and Aragorn seems to mostly go along with him. Aragorn's growth into his role as King Elessar instead comes with his healing of Eowyn and Faramir, and with his daring trek through the Paths of the Dead. The passing of the torch of hope has to happen later, with the hobbits' return to their homes for the Scouring of the Shire, and Gandalf specifically tells them that they'll have to handle it themselves.
The Scouring of the Shire does let Tolkien complete what he has to say about pity. When the hobbits capture a disempowered Saruman who, only motivated by spite, has been destroying the Shire, Frodo makes the decision to let him live even after that, paralleling what happened with Gollum. While Scouring of the Shire tells a better story of pity, I think it tells a slightly less good story of hope than Aragorn in the Last Debate.
Although I still enjoy the pre-Bree chapters and the epilogue/Last Debate in the books, I think the movies are at least more thematically tight and tonally consistent in these areas. There are many things I enjoy more in the books, one example being Gimli, Merry, and Pippin being less farcical, but I can see how these characters being comic relief helps the management of tension in the movies. I also love Tolkien's use of language, in particular his selective archaisms from the Elves and in the libraries of Gondor, his wide vocabulary for the natural world, the way Eowyn code-switches in her desperate conversations with Aragorn, and the poems and songs scattered throughout.