Revealing mistake: In the Prologue, in the shot where Elendil is thrown aside by Sauron and hits a rock and then falls to the ground, you can see his shoulder armor bend, which metal armor should not do. This is because the filmmakers used rubber armor for fighting scenes. This is confirmed on the DVD audio commentary.
Continuity mistake: When Pippin says "What about second breakfast?" he's crouched over by the pony, a bit away from Merry. Aragorn says a line and the camera goes back to the Hobbits. Now Pippin is further from the pony standing right by Merry, not crouched at all. He couldn't have moved that far during Aragorn's line and be standing so naturally.
Continuity mistake: When Merry is collecting firewood at Amon Hen he has a visible breath. None of the other characters do however.
Other mistake: When Frodo is riding along with Gandalf the Grey as they pass the kids in the cart, we see a shot of the path they are riding along that shows an automobile tire tread in it.
Continuity mistake: Not only do the arms on the two large statutes at Amon Hen change, the statutes themselves switch places. When we first see them, the statute of the younger man has a winged helmet and is on the left and the older with the smooth helmet is on the right. In the shot once the boats have gone past, the winged helmet statute is on the right and the smooth helmet statute is on the left.
Continuity mistake: When we see Frodo and Sam at the night party, there's a girl with curly hair behind them. Depending on the angle she is either dancing with friends, or walking alone. This changes back and forth.
Answer: The temptation of the Ring is directly proportional to the power and ambition of the bearer. To someone like Gandalf - a mighty wizard who wants to save the world - the temptation would, over time, prove to be too much, and he's realistic enough to understand that about himself. With an ordinary hobbit who only wants a nice meal and some peace and quiet, the Ring has a lot less to work with.