Other mistake: Before Frodo and Sam catch Gollum we see him climbing down the cliff. The moon is directly behind him and should be the only light, yet he reflects the light on places that should be completly dark. (00:09:15)
Other mistake: In this scene, Legolas says, "The Uruks turn Northeast. They're taking the Hobbits to Isengard." This is a mistake because Aragorn, Gimli, and Legolas are chasing the Uruks on a westerly course towards Isengard. If the Uruks had turned Northeast, they would have almost doubled back on the trio. (This is in the "Special Extended Edition" DVD.) The Uruks were taking the closest way to Isengard, across terrain, not traveling by common roads. In the book they traveled without rest. There's no reason to believe they would have diverted from their Western course to Northeast temporarily, that would have slowed them down. (00:19:00)
Other mistake: When Eomer is mounting his horse after giving Aragorn, Legolas and Gimli the two horses, you can see his sword falling out of its sheath. (00:35:00)
Other mistake: It seems as though the Elves at the battle of Helm's Deep aren't quite as coordinated and elegant as Elves are rumoured to be. During the Helm's Deep battle scene, Aragorn is shouting commands to the Elven archers. In the close up shot of Aragorn while it's still raining, look to the left of Aragorn, there is an Elf who gets hit in the head by the quivers on the back of the Elf next to him. (01:04:50)
Other mistake: When Aragorn, Legolas, Gimli, and Gandalf are at Edoras and are first entering Theoden's hall, watch the right side of the screen as Wormtongue says, "I told you to take the wizard's staff..." Legolas delivers an uppercut to a guard, but his arm goes way above his head. Furthermore, the punching sound effect is not heard until his arm is way over his head. Legolas is as tall as the guards and his fist would have stopped as soon as it hit the guard's face. There is no way the guard can be that tall. (01:19:05)
Other mistake: Just as Gandalf and the Rohirrim are riding into the Uruk-hai and they clash, go slowly, frame by frame. Legolas will be on the right of the screen. Keep going frame by frame. You'll soon see a few pretty clear frames of Arwen with her sword, on her white horse. She's sitting to the right of Legolas. Her long hair is pulled back into a braid. She's wearing pink sleeves. She's the only one on the screen with that color, so you can't miss it (slow motion can be helpful but is unnecessary). The official reason for editing out Arwen (given on the DVD extras) was that during filming Peter Jackson intentionally leaked an image of Arwen at Helm's Deep online to gauge fan reactions, then backpedalled over her presence after overwhelmingly negative fan reactions. (01:30:40)
Other mistake: When Faramir and his men are looking at the map of Middle-Earth in Henneth Annûn, you can see that the plain of Dagorlad is actually spelled Dagorland on the map. An odd mistake, especially considering the fact that they got it right when they spoke of it in the first movie. (01:45:20)
Other mistake: John Noble's name is misspelled in its second appearance in the SEV credits. It is correct in its initial appearance (the fades with the fancy font). However, in the short list of where the actors are listed with their character's name, it is spelled "John Nogle." (01:50:50)
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)
Other mistake: At the very end of the film, where Sam, Frodo and Gollum are approaching Mordor after leaving Osgiliath, it pans up over the mountains to reveal Orodruin (Mt. Doom) in the foreground to the right and The Tower of Barad-Dur in the background to the left. According to Tolkien's maps, as seen in the prologue to the Fellowship of the Ring, if they were approaching Minas Morgul from Osgiliath, Orodruin would be to the left and Barad-Dur to the right.
Other mistake: When Aragorn & Gimli are being pulled up the wall after defending the gate at Helms Deep, Gimli is a puppet (A very good puppet though.) when you see him in Aragorn's arms just before they get pulled over the top by Legolas, only a quick shot but still visible.
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?
Other mistake: When the refugees from Edoras are on the move to Helm's Deep, against the mountain range backdrop, they are going the wrong direction for a mountain range viewed from the north, where Rohan and camera are presumed to be. They should go from left (east) from Edoras to right (west) to Helm's Deep, but instead we see them going right to left, i.e. west to east. This occurs in two long-range wide-field shots.
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.