Continuity mistake: Faramir hits one of the Haradrim riding the Mûmak with an arrow and he falls. As he hits the ground, in the background are Frodo and Sam. Sam's cloak is not draped over his left shoulder in this shot. In the very next close-up, Sam's cloak is draped over both shoulders. (01:44:55)
Continuity mistake: Near the Black Gate, when two of the men from the marching army are walking towards Frodo and Sam, the decorative scarves at their necks change positions and are hanging differently, than when they turn and walk away from the 'rock'. It is not windblown. (01:05:05)
Revealing mistake: Soon after the Uruk-hai breach the wall with the ladders, Aragorn uses his sword to slice the leg off of one of them. In the wide shot, the Uruk has leg guards on, but in the close-up of the leg being cut off at the knee, no guards are on that leg. Also, visible in the close-up, it is obvious that the 'leg' is prepped to be cut in half. (01:05:30)
Revealing mistake: During the destruction at Isengard, Treebeard says, "Break the dam. Release the river." While he says that, there is a close-up of the dam wall as the Ent pulls away the wood blockades. We see the water starting to spurt between certain stones. Then there is a wide shot of the area, and in this shot, the water does not start to spurt between the same stones or in the same manner as in the previous close-up. (01:31:50)
Continuity mistake: In Osgiliath, when Sam says, "I know. It's all wrong. By rights we shouldn't even be here," he is standing up right next to the wall. In the next shot, of Sam's back, he is standing a few feet away from the wall. In the next shot, he's next to the wall, and then starts to walk to the archway, for his 'Sam' speach. (01:34:55)
Continuity mistake: When the Uruk-hai slam into the chamber door with the battering ram, just before Theoden yells, "Forth Eorlingas!" we can see the damage done to the wood on the right door on the outside. As the Uruks start to run in, we also see some different damage to the metal trim on that door. However, as Theoden rides past that door on his horse, the damage to the door is gone! (01:28:35)
Continuity mistake: Frodo lies down to look at the Southrons. Sam and Gollum come close, and crouch alongside him. In these first two shots, the ground beneath them is uneven, has patches of green grass, bramble, and some small green leaves near Frodo's fingers. After a cutaway, in the next shot of Frodo, Sam and Gollum, they are lying on even ground with yellow dry flat grass. Then Sam says, "No one at home would believe this," and the ground is now the same as in the first shot. (01:43:00)
Revealing mistake: During the Warg battle, as Aragorn slices off the head of a Warg rider, in the background on the hill there are two Rohan riders on horseback, one on a brown horse, one on a white horse, riding down the hill chasing after Orcs on foot. There are many shots after, including Aragorn throwing the spear into the Warg saving Gimli, and Theoden stabbing the Warg in his mouth. The next shots of Aragorn are when he stabs a Warg rider. The background plate is the same hill and the same two Rohan riders on horseback are still riding down the hill chasing after the Orcs on foot. Obviously these two pieces of footage were shot consecutively and were cut and separated, and other various shots were inserted in between them. (00:11:35 - 00:12:20)
Continuity mistake: Frodo, Sam and Gollum are resting at the Dead Marshes. When the camera is in front of Frodo the sunlight is on Frodo's left side and on Sam's face. However, when the camera angle is behind Frodo's right shoulder the sunlight is on Frodo's right side. The right shoulder shot is shown six times and it goes back and forth between both shots. (00:44:00)
Audio problem: When Gimli says the line "Helms Deep! They flee to the mountains when they should stand and fight! Who will defend them if not their king?", Legolas, Aragorn, Gimli and Gandalf are walking towards the stables. Gimli's lips are not mouthing any of the lines. This is because it is the scale double of Gimli walking with the three other actors and John Rhys-Davies added the dialogue in ADR later.
Continuity mistake: When Boromir first goes over and hugs Denethor, Boromir puts both of his arms under both of Denethor's arms for the hug, as seen from Faramir's POV. In the next shot, from behind Denethor, we now see Boromir's left arm is on Denethor's right shoulder and Boromir's right arm is under Denethor's left arm. Extended DVD. (00:33:25)
Continuity mistake: When Aragorn first arrives in Helm's Deep and is confronted by Legolas, there are three close-up shots of Legolas saying, "Le abdollen" (you're late), "You look terrible," and then smiling. The bow visible behind his right shoulder appears to be his old Mirkwood bow. It is dark golden-brown, smooth, with a black bowstring. The one Galadriel gave him is the same color as his hair, elaborately carved, and has a light-colored bowstring. (for comparison check 1:43:20, where they are in the armory, and the bow should look even darker... but the white bowstring's visible there.) Continuity is screwed up as well, because in the shots between the close-ups of this conversation, where both Aragorn and Legolas are shown, the oddly-carved ends of his Lorien-bow are visible. (01:37:15)
Continuity mistake: Gollum's hands are bound behind his back, during Sam's speech. After Sam's speech, Faramir walks over and while he's speaking to Frodo and Sam, Gollum is shown rubbing his hands. Faramir has not yet said, "Then it is forfeit. Release them." Gollum's ties would not yet have been cut. As explained on the commentary, they needed shots of Gollum for this sequence, so they pinched them from other moments, and inserted them here. (01:35:15 - 01:37:45)
Continuity mistake: When Legolas places the Evenstar into the palm of Aragorn's hand, the chain is all coiled up. Yet when Aragorn opens up his hand, the Evenstar and the chain are lying in his palm neatly and flat. Some of these shots with Aragorn and Legolas were filmed over two and half years apart. (00:45:40)
Continuity mistake: Standing in Fangorn Forest, Legolas says, "The trees are speaking to each other," and then Aragorn says, "Gimli. Lower your axe." In the wide shot as Aragorn says, "Gimli," the trees behind and to the left of Gimli are different in this first shot then in the very next shot when he continues, "Lower your axe." The trees in the second shot are also seen when Gimli first draws his axe. (00:51:20)
Revealing mistake: In the two scenes, when Frodo and Sam are both up on the rocks and later, down below, hiding out of view, the Black Gate is in front of them. Earlier, is the view inside the gate. The two Trolls are pulling and pushing the wood 'pulley', to open the gate, while walking on the tall ramp that curves around. When the gate is open, that ramp should be in full view, from the angle that Frodo and Sam are positioned, but it is not. (01:03:55 - 01:05:55)
Continuity mistake: When Faramir and his men are looking at the map of Middle-Earth in Henneth Annûn, Minas Morgul is not shown anywhere on map. It should be at the end of the road leading east out of Osgiliath. This leads us to believe there's an non-fortified road leading straight into Mordor. (00:28:10)
Continuity mistake: Faramir captures Sam and Frodo. Faramir goes over to the dead body, and says, "His sense of duty was no less than yours, I deem," and rolls over one of the Haradrim with his foot. The dead man is lying with his face towards the ground. In the earlier shot, when he fell off the Mûmak and landed behind Frodo and Sam, the right side of his face is lying on the ground. It is the same man in both shots. (01:44:55 - 01:45:40)
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.
Continuity mistake: When the Uruk-hai run through the rock canyon carrying Merry and Pippin, it's obviously the same rock canyon that Aragorn, Legolas and Gimli run through when Aragorn says, "Not idly do the leaves of Lorien fall." However, when the Uruks run through it, many false rock formations and extra running footage were added which not only changes the appearance of part of the canyon between the two shots, but also makes it seem like a longer distance through the same canyon. (00:14:40 - 00:18:00)
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.