Other mistake: Before the battle of Helm's Deep, all of the Elves are positioned along the top of the wall. Their waists are right up against the shorter wall to protect them somewhat from incoming crossbow bolts. The Uruk-hai are heard arriving from a distance and Aragorn walks out a few feet in front of them to survey the battlefield. How could he walk out in front of them when they were positioned right up against the edge of the shorter wall?
Continuity mistake: While Sam is cooking the rabbit stew, Frodo goes to his pack lying beside the broken stone statue to take out the plates. In Frodo's close-ups the trees and bushes near the statue are positioned differently and some are even different than the trees and bushes in the wide shots. (01:41:55)
Continuity mistake: During the battle of Helm's Deep, a line of Uruk-hai with shields and a battering ram march up the causeway to the Keep. Then the wall below is breached, and in one shot of the wall exploding you can see, off to the right, that the Uruk-hai have reached the top of the causeway and are just standing there. Then in the close-up of the Uruk-hai on the causeway, they haven't reached the top yet. (01:09:10)
Continuity mistake: As Faramir recalls Boromir's speech, the Horn of Gondor is slung over Boromir's shoulder. In the shots from behind, the Horn is high up on his back, next to his cloak. In the shots from the front, the Horn is much lower at his hip, under his cloak. (00:31:55)
Continuity mistake: In the scene just after the Riders of Rohan give horses to Aragorn, Gimli and Legolas, you see them riding away. Two riders at the back of the group are riding white horses. Yet in the next shot, there are only riders on black horses at the back of the group. (00:35:20)
Revealing mistake: When the camera moves into the "Seeing Stone" and pans up to reveal "the great eye," the entire tower was a miniature. It's not that noticeable, but what really makes it a mistake is the cotton balls used to make smoke on the smaller towers leading up to the big one. (It sticks out so obviously that one must ask, why did they CG the orcs at the bottom of the miniature, but use cotton balls for the smoke instead of CGing it?)
Other mistake: In the Fan Club Credits (Extended DVD), Cate Blanchett's name is spelled wrong. Her surname is spelled 'Blanchette' in the fan club credits. (01:53:25)
Continuity mistake: When Legolas and another Elf are dragging Gimli back to the Keep, in the front view the Elf's quiver is empty. Yet in the next shot from behind, the Elf has many arrows in the quiver on his back. The arrows would be seen from the front, because they are way above his shoulder in the back shot. (01:13:40)
Continuity mistake: In the extended DVD, when Gimli quips, "That's not bad for a pointy-eared Elvish princeling!" the back shot of Legolas shows him 'toying' with his bow, but in the front shot, his hands are apart holding the bow. (01:39:00)
Deliberate mistake: In the first close-up shot of the scene, Aragorn is walking and talking to 'Legolas'. Legolas' back is shown, as Aragorn is speaking to him. At a certain point, 'Legolas' turns his head to his left and in this close-up it is very clear that it is not Orlando Bloom, but his body double, Morgan. Extended DVD. (00:50:50)
Continuity mistake: At the Dead Marshes after Gollum rescues Frodo, in the next opening shot while Sam sleeps we see the pot and pans lying on the ground near Sam's head. When he hears the Nazgul he wakes up and some of those things lying near his head earlier are not there in this shot. This has nothing to do with camera angle. (00:47:15)
Continuity mistake: On the way to the old sewer in Osgiliath, in the first shot, we see Faramir leading Frodo and Sam with Gollum in the rear. In the next two shots, Gollum is directly behind Faramir now, with the two Hobbits reaching the entrance first. Extended DVD. (01:41:35)
Deliberate mistake: When Faramir is about to release Frodo and Sam, not only are the Elven brooches the wrong way round, as someone else has noted, but the whole scene is flipped. You can tell because the leather straps that hold their baggage are the wrong way round as well. Faramir's shots go from normal to flipped a few times, when he walks over, kneels and then rises. (02:40:30)
Continuity mistake: In Osgiliath, before Sam's speech, in the wide shot showing Sam's back, the large fry pan hangs down, below the small fry pan, on Sam's back. In the next shot, in front of Sam, the large fry pan is seen sticking out, just behind Sam's right shoulder. (01:35:10)
Continuity mistake: When we first see Legolas on the wall with Gimli, Legolas is gripping his bow, left fist over right fist. Next shot as Aragorn walks over, his left hand grips the bow lower but his right hand is gripping the front of his quiver strap. Next shot, Aragorn is beside him, his left hand is much higher on the bow now. As Aragorn leaves, Legolas' left hand is lower on the bow again and his right hand is again gripping the front of his quiver strap. (01:00:25)
Continuity mistake: Just before the trees start killing the Uruk-hai, we see the backs of Gandalf and the rest of the heroes on horseback as they're watching the forest. Gandalf is holding his staff up in his right hand. Next close-up shot, he's lowering the staff in his left hand. Extended DVD. (01:38:20)
Continuity mistake: When Frodo is pulled out of the swamp by Gollum, Frodo's hair is dripping wet, then not as wet in the next shot, then his hair is matted on his forehead and then off his face. (00:46:55)
Continuity mistake: In the extended DVD, right before Faramir lets Frodo, Sam, and Gollum go, he grabs Gollum around the neck. The problem is, he grabs Gollum from behind, so his hand is on the back of Gollum's neck. He then pushes Gollum up against the wall, but suddenly his hand is in the front. I've watched this closely, and there isn't enough time for Faramir to twist his hand around. (01:42:20)
Revealing mistake: When Sam is running up the steps to stop Frodo from putting on the Ring in Osgiliath, you can see when he puts his right foot on the stair the "fake foot" toes slam into the stair and bend to a point which is impossible to achieve with a real foot.
Continuity mistake: At the flooded Isengard, the first shots of Pip and Merry are in front of Saruman's storeroom. Floating in the water, directly behind them are a large barrel and a large crate. In the shots of them turning to enter the storeroom, the floating barrel and crate are gone. Extended DVD. (01:40:15)
Answer: The movie makes it clear that Gandalf has lost some of his memory and his personality has changed. Gandalf actually did die and pass over to the "other side", as it were but was sent back to complete his task. However, he was sent back as a similar, but different entity. Gandalf the White does not have all the memories of Gandalf the Grey, at least at first. He doesn't even remember that he used to be called Gandalf the Grey until someone points this out to him. It is implied that Gandalf doesn't even remember Sam until Aragorn mentions him. Gandalf then searches his memories and remembers who Sam is and his eventual importance to Frodo's quest. When he finally remembers this, he is pleased that Sam went with Frodo, as he will play a crucial role near the end of the journey.