Saruman
You want another one? In the movie, Saruman has decided to throw in with Sauron. He makes this big speech to Gandalf about how Sauron is certainly going to win, and the only hope is to join him.
This is completely different from Saruman's motivation in the book. There, Saruman has failed the test that Galadriel passed ("I pass the test; I will diminish, and go into the West, and remain Galadriel"). He has decided to sieze power for himself, and is desparately in search of the Ring toward this end. He is a third power, standing against both Sauron and "The West" (Aragorn and company). He would never even consider being subservient to someone else.
Why is this significant? It supports Tolkien's theme of the limits and dangers of power. Saruman was motivated by good originally, but he also lusted after power; he was the strongest of the wizards, and the leader of the council set up in opposition to Sauron. He believed that the only way to accomplish his ends was to build his power. This corrupted him. Gandalf takes a different approach, trusting to the meek and to fate. Once again, we see the Christian aspects: the meek shall inherit the Earth, and only by the grace of God do you enter Heaven.
In the movie, all this is lost, and Saruman becomes a chicken, giving in to Sauron because things look tough. While that is a legitimate kind of character to present in a story, I think the character Tolkien presented is more compelling: a fallen hero, seduced into evil, nearly accidentally, by his singleminded fixation on control.
Galadriel (gifts, Gimli)
As an aside, I find it particularly impressive that Galadriel passed the test (I know, I find everything about Galadriel impressive). The reason she left Valinor and came to MiddleEarth in the first place was to set up her own kingdom; a sort of "better to rule in hell than serve in heaven" kind of thing. Frodo's offer of the Ring would have given her the power to acheive what she'd spent the last <mumble>thousand years working toward, what she'd defied the Gods to pursue.When she turned it down, it was the repentence of the last of the original sinners among the Elves. It may be the most important moment in the book, in terms of the story's placement in the Tolkien mythology: it's the end of the long rebellion of the Elves.
Another Galadriel moment cut from the movie is related here. As everyone is leaving Lorien (in the book they spend a month there, incidentally; in the movie, they spend the night), Galadriel gives them gifts. Frodo gets the "phial," as in the movie, but in the book all the others get things too. When she gets to Gimli, the Dwarf, she has nothing, so she asks him what he wants. He's completely in love with her by this time, and he says he asks for nothing, but if he had to name his heart's desire, it would be a lock of her hair. She laughs, and gives it to him.
Long ago, Feanor, the greatest elf in history, had also asked this of Galadriel. She turned him down. Maybe she turned him down because she didn't like him (which she didn't; he wasn't a likeable guy...he knew very well that he was the greatest elf in history), but I think it's more likely that she turned him down because she was "Galadriel, dammit, and what kind of cheek do you have, asking for a lock of my hair!" Pride goeth before a fall. Once Galadriel has refused the Ring, she has made the final step...she has conquered her pride. She can now give a lock of her hair to a mangy little Dwarf, and feel good about it.
Of course, none of that is in the movie.
Gandalf & Moria
Here's another one. In the book, Gandalf wants to go through Moria. Before they even leave Rivendell, he suggests it to Aragorn. Aragorn is against it, and talks Gandalf out of it, although they continue to argue all the way to Caradhros. This seems important in two ways: First, Aragorn wins the argument--Gandalf defers to Aragorn, because he has declared himself King at Rivendell, and this is all his fight now. Aragorn is, after all, the main character in the story of the War of the Ring. This is our first indication of that, and it leads up to the departure from Lorien, discussed above.
Second, it seems significant that Gandalf wants to go to Moria, and he dies there. Does he know, and go anyway? (As an aside, in the bookhe doesn't know there's a Balrog in Moria ahead of time. He suspects something is there, but he doesn't know it's a Balrog. And Legolas is the one who recognizes what it is when it attacks.) Or is this a failing in his wisdom? Or does he know, knowing he will die, and know his death is important in some way...perhaps to put Aragorn in charge? There are arguments all 3 ways, all of which contribute to the themes of the book--the limitations of Power, the importance of fate and the little people. Those don't seem like "Page 56" things to me.
Aragorn after Gandalf's death
You seem like a 'what about page 56' kinda guy....
I don't think it's stuff like that. Here's an example, in my own defense. In the book, after they leave Lorien and get in the boats, Aragorn has this huge Hamlet-esque crisis. He's in charge now, for the first time in his life, and he can't decided whether to go to Minas Tirith and claim his kinship (and Arwen) or to go East with Frodo to try to destroy the ring. This is actually why they get boats from Galadriel--Aragorn confesses that he has no idea which side of the river they want to be on, so Galadriel gives him boats so he doesn't have to decide yet.
In the end, Aragorn doesn't ever really decide. Frodo slips away, while invisible, and thus fate makes the decision for him. This is in support of Tolkien's big theme: no matter how great or powerful noble, you can't win if fate is against you. (I suspect it's a Christian message, judging from the way it's presented in the long story of the Silmarils from Feanor to Earendil)
In the movie, Aragorn show no indecision at all. That seems like a significant failing to me.