Continuity mistake: In the scene where Picard, Worf and Hawk are outside the ship and want to separate the transmitter dish from the hull, Picard must move a kind of tube out of an console and must turn it from a low to a high position. In one shot the tube is in the high position, in the next shot it is in the low position and then Picard pulls it out and turns it in the high position.
Continuity mistake: When Lilly and Jean Luc are arguing in the captain's ready room Jean Luc breaks the glass cabinet holding the gold models of the previous enterprises with one of the Phaser rifles. It only causes the gold model of the Enterprise D to spin 90 degrees on its hook. The next shot both the gold models of the Enterprise C and D are broken in half.
Factual error: In the scene after Zefram's ship goes to warp, when the ship is being turned around to return home, Cochrane remarks "Is that the Earth? It's so...small." He wouldn't have been able to see it, provided the distance the ship just flew. Warp 1 is the speed of light and in a 1 minute flight, the Phoenix would have travelled over 10 million miles - at that distance the earth would just appear like a bright star.
Continuity mistake: As the Enterprise crew walk towards Engineering for the first time, Worf senses the Borg awakening and yells "Ready phasers" before killing a Borg. As that particular drone awakens, we can see it has a needle-like tool coming out of its eye sensor. But when Worf smashes it with his rifle, it has a blue lit-up eye sensor. Furthermore, this drone is seen many times during the following fight sequence after Worf has apparently killed it.
Factual error: In the scene where Troi is drunk and Riker asks if Cochrane can accept the truth, it is night. Then the scene switches to the Enterprise orbiting Earth over north America and it is apparently daytime because the sun is shining over north America, where the movie takes place.
Continuity mistake: During the Dixon Hill holodeck scene, the room was full of "people" running around while Picard fired the machine gun at the Borg. Yet a few seconds later the room was totally empty. There wasn't enough time for everybody to have abandoned the room. And if it had been a security feature, the entire simulation should have ended, instead of just disabling all the people.
Continuity mistake: The number of Borg working on the transmitter dish changes and they move with the speed of light. While Picard explains how they should disable the dish, a shot show that 6 Borg are working on it. They disable 3 in various ways and 1 walks off with Hawk, leaving 2. However, as the disk is being released from the ship there are 3 Borg on it. When Picard disabled his "maglock servo" he avoids capture by a Borg by demagnetizing his boots and flying over it. When he lands at Hawk's earlier position, the Borg is still standing where he left, however a shot seconds later shows the Borg back on the dish.