The Midgewater Marshes
Aragorn leading the Hobbits through the Midgewater Marshes is important for a couple reasons. One, it shows how the Ranger was able to protect and feed the Hobbits on their trip to Rivendell. Two, Aragorn sings the Song of Beren and LÃºthien. Their story mirrors that of Aragorn and Arwen, and it was so important to Tolkien that Beren is carved on his tombstone, and LÃºthien on his wife’s.
Where There’s a Whip There’s a Way
That song from the Rankin-Bass animated version of The Return of the King didn’t make it into Jackson’s adaptation, but we thankfully got to see Frodo and Sam joining the column of Orcs. In the theatrical cut, they leave Cirith Ungol and then arrive at the base of Mount Doom. We didn’t really see them crossing the Plains of Gorgoroth much, and with them joining the Orcs and then shedding their disguises in the Extended Edition; that bit is restored.
There is a really cool bit added to Moria in the Extended Edition where Gandalf explains that the wealth of Moria was in Mithril. He mentions how Bilbo had a Mithril coat (which Frodo is secretly wearing) that Thorin gave him. That “Kingly gift” happens at the end of The Hobbit, and if Jackson doesn’t include it in his adaptation; the movies will then have a plot hole the size of Leia remembering her mother.
In the book, Faramir is famous for saying “I would not take this thing, if it lay by the highway. Not were Minas Tirith falling in ruin and I alone could save her, so, using the weapon of the Dark Lord for her good and my glory. No, I do not wish for such triumphs.” Naturally his character was changed in the movie to the ire of purists, but as Jackson explained characters in a film have to grow and have an arc; and the book version of Faramir didn’t have that. In the Extended Edition, Faramir does get a bit of his book personality back when he gives a similar speech to Denethor.