Audio problem: When Sam, Frodo, and Gollum are in the forest of Ithilien, when Sam says "Can't you hear yourself, don't you know who you sound like?" the shots change and you can see Sam mouthing the words again. (01:38:30)
Audio problem: When the Ents are attacking Isengard, just after the river has been released. Merry calls to Pippin "hold on!", but if you look at his lips, he says "Hang on!" The lip-syncing doesn't match. (02:35:25)
Audio problem: First there is a wide shot of the Dead Marshes and Gollum says, "Soft and quick as shadows we must be." Then in the next shot as the camera starts to pan towards Sam, he begins to say, "I hate this place, it's too quiet..." When the camera first shows Sam, his mouth is not in sync with these particular words, "I hate this place," the words originally said on camera were changed during ADR. Extended DVD. (00:42:45)
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.
Audio problem: On the Deeping Wall, Theoden says, "Draw all our forces behind the wall..." but Theoden's lips are moving before we hear him say those words. (00:17:10)
Audio problem: When Gandalf rides off to find Eomer, Shadowfax isn't wearing any sort of harness. But just as Gandalf rides out of the stable past Legolas, you can hear a jingle, like a bridle would make. (01:30:20)
Audio problem: When Gollum starts leading the hobbits, just after they let him go, he slips slightly on one slope and there is a sound of stones sliding down. But no stones are dislodged.
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.