9. Changing Saruman's Relationship With Sauron
Christopher Lee played the evil wizard Saruman with such captivating affect that no one doubted the power and menace of his character. He's presented as a substantial obstacle for our heroes, particuarlaly in the second film, but Peter Jackson stripped the character of much of his nuance.
In the films he's presented as just another servant of the Dark Lord; someone who is corrupted by evil, and simply willing to do the bidding of Sauron. But in the books it's very different. Saruman becomes so powerful and arrogant that he believes he could overthrow Sauron himself. He asks Gandalf not to join with Sauron but to help him replace the Dark Lord and rule in his stead.
This would have set up a whole new dynamic in the films. Saruman's motivation for attacking Rohan is in part due to Gandalf's refusal to help him find the One Ring. We also never got to see Saruman's interactions with the Nazgûl. In the books they visit him in search of knowledge, but Saruman denies them entry to Isengard, instead sending them after Gandalf.
This results in another moment that would have been an onscreen spectacle. When the hobbits arrive at Weathertop, Aragorn finds evidence that Gandalf had been there just days before. Gandalf later reveals he was forced to battle the Nazgûl before fleeing. That's a scene we would have loved to see.