Continuity mistake: When the Warg raises his head from his kill, he spots Gimli and he growls, showing his blood stained teeth. There are spikes on both sides of the harness around his neck. When he charges toward Gimli, Legolas kills him and he falls right in front of Gimli. There are no spikes on the harness, nor are his teeth stained with blood as before. (00:11:20)
Continuity mistake: Just before Legolas surfs the shield down the stairs, he is holding the shield in his right hand and his bow in his left hand, which is down. In this shot, he starts to toss the shield down onto the ground, still clutching the bow in his left hand. In the next consecutive wide shot, the shield is still in mid-air, and as he is starting to run towards the moving shield, Legolas does not have the bow in his left hand now. He did not throw the bow over his shoulder. The next shot is a close-up of his feet on the shield, just starting to come to the stairs. In the next wide shot, as he is starting to surf at the top of the stairs, Legolas has the bow in his left hand again and he is lets loose the arrows at the Uruk-hai. (01:11:00)
Continuity mistake: When Theoden is mourning his son's death at his tomb, Gandalf is holding his staff to his left when shown from the back. When shown close-up from the front, however, he is holding his staff to his right. (01:04:20 - 01:05:55)
Continuity mistake: When Saruman is telling the Orc to send out his Warg riders, in the close-up, Saruman lets his staff hit the ground and then in the shot of his back, his staff goes down again and it is in the opposite hand. (01:36:40)
Continuity mistake: Upon entering the Golden Hall, when Aragorn, Gimli, Legolas and Gandalf first arrive at Edoras, watch Gandalf's staff. In the shots from behind, Gandalf uses the staff as a walking stick, but the shots taken from the front, show him keeping the staff at his side, perpendicular to his body. (01:17:50)
Trivia: At the battle of Helm's Deep, when the Uruk-Hai are storming the gate, we see a black-haired man throwing a spear. The man is director Peter Jackson in a cameo.
Continuity mistake: When speaking of Merry and Pippin, a certain Orc says, "What about them? They're fresh." This Orc's face has very distinctive marbled skin coloring. After Grishnak says, "What about their legs? They don't need those..." when that other Orc says, "Just a mouthful," before Uglúk cuts off his head, the skin coloring on his face is quite different. (These were shot at different times.)
Continuity mistake: When Sam, Frodo and Gollum arrive at the Black Gate, they hide at the top of the cliff looking down on the soldiers approaching Mordor. As the ground crumbles beneath Sam, the soldiers are some distance from the Gate's entrance, the shot switches as he falls then quickly resumes its original position. Yet the soldiers have now entered the Gate, all within the space of a split second. (01:04:30)
Continuity mistake: When Aragorn is dreaming of Arwen, they stand and kiss. In the close ups, his hands are on her cheeks and jaw, but in the wider shot his hands are on her waist. (00:05:35)
Continuity mistake: Faramir and his men capture Frodo and Sam, and Sam is thrown to the ground. In these shots, Sam's cloak is lying straight under him, as can be seen when he says, "Wait. We're innocent travelers." When one of Faramir's men grabs Sam's shirt to pull him up, his cloak is lying on the ground, up near his shoulder out to his right. Sam has not moved, Faramir's man had a blade to his throat. (01:45:15)
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.