Examples? I recall the films omitting some stuff for the sake of time but I don't recall any significant deviations from the books.
Hoo boy. I used to mod the countingdown.com LOTR boards. Here's a bunch that I remember off the top of my head that got fanboys in a tizzy.
- No Tom Bombadil scene, and no Barrow-wight scene.
- No Scouring of the Shire
- Faramir changes; in the books he's calm, assists Frodo and Sam, and even Gollum after he was captured, and wasn't tempted by the ring and never took the hobbits as prisoners.
- Arwen in place of Glorfindel (taking Frodo to Rivendell.)
- No Elves were in the book in Helm's Deep. The LOTR-faithful were in a complete mess when they heard rumors about Arwen being at Helm's Deep as a "warrior princess" role (whih was filmed.)
- Aragorn accepts his role as the heir much earlier than ROTK in the book. He's also in general much more noble and calm then the way Viggo portrayed him.
- Denethor is portrayed as a psychopath in the movies where in the books the Palantir (and Sauron) has worked to fill his head with tons of Dark-lord-propaganda, him being really a wise man.
- Theoden isn't "possessed" in the books, but rather just manipulated by Wormtongue.
- TTT movie added the whole warg attack and Aragorn going over the cliff and catching up later at Helm's Deep.
- Sam never was sent away by Frodo (in TTT) as in the movie.
- The Entmoot was pretty different; the Ents were pissed all to hell about Saruman's cutting of the forest and wanted to meet and go to war, but in the movie they decide not to until they stumble upon the decimation.
... and many more. Some are omissions of characters (Glorfindel, Prince Imarhil, Radagast, Beregond, etc,) dialogue changes (Eowyn vs. Witch-King is a good example) while some are just subtle changes here and there, added scenes and such.