Continuity mistake: Before Gandalf makes his reappearance, when Gimli says "The air is so close in here" he has two axes at his waist, one in hand (and one on his back as seen earlier). The axe on his waist at his left has a solid head. Later, when Aragorn tells him to lower his axe, one is missing from his waist. It reappears in his hand just as Aragorn starts to draw his sword. Then, just as the three are about to attack the wizard, Gimli doesn't appear to have an axe in his left hand. Yet, in the subsequent cut, he throws one with his left hand. Then, when Gandalf reveals himself and Gimli bows his head, the solid-bladed axe appears to have moved to the right-hand side of Gimli's waist (as they cower in Gandalf's brightness). (00:39:55 - 00:42:00)
Revealing mistake: Legolas runs down the hill to kill the Orc scout. As he is slicing the Orc's throat, in the close-up of Legolas' blade, one frame shows the position of the sharp edge right up against the Orc's throat, but the next frame shows that the sharp edge of the blade was turned away from the Orc's neck, just before the slicing, so as not to harm the Orc actor. This can only be seen frame by frame. (00:09:10)
Continuity mistake: In most scenes with Gollum, he has a small pouch attached to his narrow belt over his right hip. Early in the scenes where Gollum is leading the Hobbits through the bog, you don't see the pouch. Both in the preceding and following shots he had the pouch. (00:41:40)
Revealing mistake: In some shots you can tell that they used children or small people, for Frodo and Sam. Mostly, you can tell when they're shown from behind, in the company of tall men. You can also tell, from the slightly different hairstyles.
Continuity mistake: When the three companions first meet Gandalf, after Aragorn drops his sword, Gimli lowers his axe and holds it by his right hand. When the shot changes, Gimli holds it by two hands and in a ready position. The same thing with Legolas, his bow changes position, and Aragorn with his legs. (00:52:50)
Revealing mistake: When Gollum brings the two rabbits, he drops them onto Frodo's lap, and Frodo is leaning up against one of the broken stone structures. After Frodo opens his eyes, the stone structure behind him vibrates when he moves a couple of times. This structure should be way too heavy to jiggle when Frodo moves. (01:41:05)
Revealing mistake: When Aragorn tracks Merry and Pippin near the beginning of the movie, after meeting Eomer, the trio stops at the border of Fangorn forest and you can see in this closeup that it's the scale double Brett portraying Gimli, not John Rhys-Davies.
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: At Osgiliath, when Frodo first brings Sting to Sam's throat, in the first close-up the chain is not caught up on the vest button. When Frodo falls back and drops Sting, the Ring's chain is hooked up on the button. The shot changes to Sam then quickly back to Frodo, and Frodo says, "I can't do this Sam" and the chain is no longer hung up on the button. (01:34:50)
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)
Revealing mistake: When Aragorn saves Gimli from the warg, the spear he throws is CG. He barely tosses the thing with little more thrust than his forearm can allow, and it pierces straight through the warg's tough hide. (00:12:10)
Continuity mistake: The flower that Theoden is holding while he is in front of his son's tomb has 8 petals, but when he drops it, it has only 6, and the flowers on the ground have only 5 petals. (01:25:30)
Deliberate mistake: In one of the first shots of Aragorn, Legolas and Gimli, we see them running on top of a hill, with the camera sweeping over them. In that scene, Legolas' quiver is leaning on his left side and Aragorn's sword is on the wrong side. This shot is flipped. (00:17:35)
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)
Continuity mistake: The Evenstar on Aragorn's neck moves from side to side during the scene with Gandalf in the stable, before the evacuation of Edoras and also at Helm's Deep when Aragorn is talking to Theoden on the walls. (01:29:30)
Continuity mistake: At the end, it shows Sam and Frodo walking through a forest. The very first shot of that scene shows Sam saying, "I wonder if we'll ever be put into songs or tales." In the background, through the trees, you can see the city of Osgiliath. Then it shows Frodo, with Sam in the background, and Frodo asks, "What?" In the next shot they are way further into the forest and you can not see Osgiliath. (01:44:40)
Continuity mistake: At the end of the film, when the heroes of the big battle are watching the lightning over Mordor, while Gandalf says the words, "The battle for Helm's Deep is over", over Legolas' right shoulder is a far view of the Deeping Wall and the Causeway. There is no break in the wall, or ladders at the wall, nor are there any bodies anywhere on the Causeway or otherwise. (01:40:13)
Continuity mistake: After Eomer gives Legolas, Aragorn and Gimli the horses and the Riders of Rohan are riding away, Arod (Legolas' horse) is being held by Legolas' right hand, but in the long shot, Arod is being held by Legolas' left hand and the horse is several feet to the left. (00:35:20)
Revealing mistake: When Faramir has his sword tip around the chain holding the ring, the blade Faramir is holding is a dead blade i.e. it has no sharp edge. (00:42:15)
Continuity mistake: Many times, when Gimli is seen with other people from a distance, it is clearly a smaller double which can be seen from the dark makeup around the eyes. This is especially noticeable when the shots are intercut with close-ups of Gimli. (00:37:55)
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.