Other mistake: It's quite convenient for the plot that Natalya knew about both Goldeneye satellites but never seemed to know where Mischa was operated from otherwise she and Bond could have gone to Cuba without having to spike Boris' location.
Revealing mistake: In the pre-title sequence, there is a shot just after Bond leaves the Russian facility while escaping. In the shot, there is a plane moving, and in the background, Bond is standing and aiming his gun at the plane. The person appearing isn't Brosnan; he is a stunt double. The stunt double's face is visible on Blu-ray. He's noticeable even without stopping the movie. (00:08:20)
Continuity mistake: When the dead admiral falls out of the closet on the boat the morning after Xenia kills him, and James Bond opens the closet door, as he's falling he has a straight face and eyes looking straight ahead, but after he lands and it shows his face, his eyes are looking upwards and he's grinning. (00:25:55)
Factual error: When Alec and Xenia escape from the train with a helicopter, that helicopter is a Robinson R22, piston engine powered. So we shouldn't hear a turbine noise but a sound like an car's engine. (01:27:00)
Continuity mistake: After Bond finds the dead Admiral, he jumps off the yacht onto a smaller boat and speeds off towards shore, but he never unties the smaller boat from the yacht. (00:25:10)
Continuity mistake: In the interrogation room, Ouromov takes Bond's gun and shoots Defense Minister Mishkin. While the gun is clearly aimed between Mishkin's eyes, he clutches his throat as he falls. (01:14:15)
Continuity mistake: When 007 and Natalia are in the plane flying over the pond and looking for the dish, Bond says "Maybe Wade was right. There is no dish." As he is saying this you can see that they are leaving the pond behind. This is obvious as you can see the trees on the same side they are flying over. Yet the missile shoots out of the pond and hits the wing while the plane is over the pond. There would not have been enough time to turn the plane round and go over the pond again. (01:37:00)
Revealing mistake: In the scene containing Q describing the operation of the "grenade" pen to Bond, Q is not looking at Bond and appears to be being cued. With careful observation you can see Q's eyes reading the cards. Also, his actions/reactions do not seem fluid but abrupt as he reads to a point where he must do an action of some sort and he performs it suddenly. Much of the scene is shot from a profile view of Q and Bond so as to reduce the obviousness of it (mostly during the longer dialog pieces for Q). (00:49:30)
Continuity mistake: When 007 escapes from Ourumov, he shoots dead several Russian soldiers (two of them from the floor), jumps through a window, and smashes through numerous walls in the tank, yet his suit when he gets into Trevelyan's train is perfectly dry-cleaned and unblemished. (01:15:15)
Continuity mistake: In the scene where Bond is escaping from the archives hall with Natalya, he jumps through a window. In the interior shot, his head breaks through the main semicircle crossbar of the window. When the shot switches to outside, the crossbar is intact along with the glass above it. After Bond's recovered, the window is visible in the top left corner and the top semicircle of glass is not there. (01:16:50)
Continuity mistake: When the Mig is falling on the antenna, it goes straight to it at a steep angle, but in the next shot, it passes through another building before and at a different angle. (00:39:00)
Revealing mistake: In the airfield outside the chemical weapon factory, James Bond hits a man with a motorcycle. He flies in the air before the impact and it's obviously a dummy. (00:08:40)
Continuity mistake: When Bond is racing Xenia at the beginning of the film, the rear Ferrari 355 badge is in one scene there, in the next missing. (00:13:40)
Revealing mistake: When the roof of the train opens so the helicopter can take off, it shakes a lot. It doesn't really look armored. (01:27:05)
Continuity mistake: When Bond and Natalya are being chased through the library by Orumov and his soldiers, Bond pulls out his belt and activates the piton with the high tensile cable. It goes into the wall just above the arched window. Before the camera angle changes to show the point of contact with the window, Bond is facing forward. As he hits the window he is suddenly facing backwards. Then as he exits the window, the camera outside shows him to be facing forwards again. (01:16:50)
Visible crew/equipment: At Q's lab, when Bond opens the car door a huge round stage light is reflected, which wans't there before in the previous opposite angle.
Continuity mistake: The first time we see St. Petersburg the amount and type of people behind the soldiers parading changes between shots.

Continuity mistake: Xenia's hair is tied up in a pony tail, but when Bond fires at the helicopter it's loose. Then it goes back to a pony tail. (01:43:00 - 01:44:01)

Revealing mistake: When Xenia dies and the helicopter is about to explode, the pilot is replaced with a very obvious stiff dummy which falls forward without control.
Revealing mistake: Bond's stunt double is incredibly visible running up the stairs at the antenna, while being shot at by 006. The double is visible again when Bond jumps down the stairs and rolls. (01:58:10)
Chosen answer: The real answer, which I believe was explained better in the game or in special features. Regardless, I did hear, is that Ourumov wasn't aiming directly at Alec's head, but to the side and shot the ground behind him. But made it look like he shot him in the head. Alec would have felt the bullet whiz by him.
Quantom X ★
That makes more sense.
Ssiscool ★
It has been a while since I watched the movie, and my copy is a VHS... but I believe if you watch very closely you can actually see the bullet hit the ground behind him with a tile breaking. I don't remember for sure if you can, but I seem to remember they did add that little effect.
Quantom X ★