Lily opened fire on Data, putting holes in his clothing. However in all following shots where Data is wearing them, no holes are present.
During Picard's briefing of the commando teams about their objectives, they arm themselves with phaser rifles. As Picard and Data approach the Borg lair, their team is holding a different kind of weapon. During the subsequent fight with the Borg, everyone is again holding the first type of phaser.
When the camera gets a shot of the holodeck doors right before Picard shoots the Borg wall with his phaser, the ID sticker on the door reads "08 HOLOSUITE 4". When the Borg force their fingers through the seam to open the door, the ID sticker now says "0820 HOLODECK 07".
When Data is captured by the Borg, they drill 2 holes in the side of his head to "find his weakness". In subsequent shots, there are no holes.
When they are showing the "maglock" handles, some of the shots show that the locks are already in the "open" position before they are turned and unlocked.
In the film's ending credits, James Cromwell's role is listed as "Cochran", when it should have been spelled "Cochrane".
When the Phoenix launches we see Cochrane, Riker and LaForge rocking back and forth in their seats. The thrust of launch is constant, and substantial, so they would not rock but rather be forced firmly and steadily back into their seats until the thrust ended. Also, their seats offer totally inadequate head support so it's quite likely all would have died from broken necks shortly after launch.
Early in the movie after Picard has decided to disregard orders and is returning to Earth we see him in his quarters looking out into space with several stars twinkling. They shouldn't be twinkling as there is no atmosphere for the light to travel through which is what causes stars to twinkle.
When Lily is shooting at Data in the missile silo, after she runs out of ammunition her gun makes several metallic clicks. This is apparently meant to indicate the weapon is empty. In reality the gun's bolt would lock open after the last round was fired. Even if she continued to pull the trigger, it would not make any such sound.
During the battle with the Borg cube, Picard orders the Defiant survivors beamed aboard the Enterprise. Transporters don't work with shields raised.
When Geordi is asking Cochrane to look at the intermix chamber blueprints, he is wearing sunglasses, even though his artificial eyes don't require protection from the sun. The sunglasses are probably needed in case a local comes looking around. Only Cochrane and Lily knew about time travelers, and Geordi's futuristic implants could blow their cover. Geordi used dark glasses for the same purpose in Time's Arrow, Part II (1992) as well.
When the away team beams down to the surface for the first time, a visible breath due to cold weather can be seen coming from Data. While it may seem strange that an android has to breathe, it's not an error - Data does have a functional respiratory system. This issue was addressed in Birthright, Part I (1993).
Cmdr Riker unplugs the jukebox when he first meets Dr Cochrane. A few minutes later Dr Cochrane appears to simply hit the jukebox to turn it back on. However, in the background during the last few lines of Riker and Troi's conversation, you can see Cochrane bend down beside the jukebox, (presumably) to plug it in. He then hits it to restart it.
How did Picard know about the weak point in the Borg cube? He had been assimilated in The Best of Both Worlds Part 1 (1990). After being rescued and disconnected from the Borg Collective in The Best of Both Worlds: Part 2 (1990), he stated that he remembered everything. Since all Borg share knowledge through the Collective, Picard learned about the design and construction of a Borg cube. Thus he knew its weak point. Also, just before he gave Data the coordinates, he heard a Borg chatter in his head (the Queen later confirmed that he wasn't imagining it), so it's possible that he intercepted a report about some kind of damage or dangerous power buildup in that section, which just turned it into the cube's Achilles heel.
In the holodeck scene, it is established that the Borg shields cannot fend off crude solid projectiles. In the deflector scene, Worf kills one with a Klingon blade weapon called a "mak'leth". Why didn't Picard simply order the crew to replicate obsolete firearms and swords to fight with? Because Borg can adapt to (almost) anything after a short while. Using old weapons would have been no more effective than reconfiguring phasers. But why did these weapons work in the first place, if Borg can adapt to them and probably encountered such weapons before? Because Borg don't use every adaptation from every weapon they have ever encountered - that would be highly impractical and inefficient. Instead, they simply re-adapt for any given mission.
After Picard has convinced Lily he is a friendly, she puts her earlier-acquired phaser in his hand. Shortly before it touches Picard's hand, the emitter lights up, indicating that Alfre Woodard (Lily) has touched the fire button of the prop, which lit up to show the post-production people where to insert a phaser beam effect.
As the Borg Queen tries to maintain her grip on Picard's leg in Engineering, she leaves a smear of metallic paint/makeup on his pants.
The Defiant is shown to be smaller than its scale compared to the Enterprise-E. The Defiant is a small ship, but it is shown at about one-third of its size when the Enterprise-E flies up next to it.
Captain Picard can be seen silently mouthing while Lily speaks, just before he smashes the display case with a phaser rifle in the observation lounge on the Enterprise.
During the battle with the Borg Cube, the Earth is shown in a last quarter phase (right half of it is dark). However, the night side is supposed to be pitch black with city lights, not the slightly darkened version of the day side with no city lights. Also, the cloud patterns across the Pacific ocean change drastically during the relatively short battle.
When Picard decides to beam down to Earth to check what damage the Borg have inflicted on Cochrane's camp, he says "computer, mid 21st century clothing" while walking to the turbolift, which contains no replicators. It seems like his quarters or his ready room would be the more suitable place for this.
According to Metamorphosis (1967), Zefram Cochran was born in 2030, but this film takes place April 5, 2063, making him 32 or 33 years old. James Cromwell was visibly older than that, in fact he was 56 years old.
When Picard is escorting Lily through the ship 49 minutes into the movie, they step through a doorway and a member of the film crew is visible in the corridor behind them, wearing a blue and white striped shirt with a radio clipped to his belt.
When Data and Picard turn a corner in a corridor before opening the hatch to deck 16, a stage light with an orange over is visible for a moment as they round the corner.
After Data pulls the manual release on deck 16 and the Borg attack, you can see extras behind them standing still.
The Enterprise is sent to patrol the edge of the neutral zone between the Federation and the Romulan Empire when the Battle of Sector 001 occurs. When Picard orders the Enterprise to return to Earth to engage the Borg, the ship seemingly gets back to Earth in a short time. However, since Star Trek maps list the Neutral Zone as being about 30 light years from Earth, it would take the Enterprise over 7 days to return to Earth at Warp 9. (When the Enterprise is ordered to the Neutral Zone, the next shot shows the ship still traveling at impulse. Is it possible they were closer to Earth and just took long range scans of the Nuetral Zone?)
When Cochrane views the Enterprise through the telescope, it is moving across the sky. When Picard and Lily look down on the earth through a force shield, the Enterprise is orbiting above Australia. For the crew to teleport between earth and the Enterprise at will, Picard would have placed the Enterprise in geosynchronous orbit directly above Montana. The Enterprise should have appeared motionless to Cochrane.
When Cochrane flies the shuttle for the first time and the Vulcans detect him, he returns back to earth and the Vulcans soon follow and land. How exactly did he land the shuttle? The rocket used was not the landing kind.
It is stated that the Enterprise is ordered not to join the fight against the Borg cube attacking Earth because, "Starfleet has every confidence in the Enterprise and her crew, they're just not sure about her captain." If the Enterprise is, as Geordi suggests, the most powerful and advanced ship in the fleet, it would make sense for her to join the fight. If Starfleet views Picard as a liability in the fight, the simple solution would be to temporarily transfer command to Commander Riker, who successfully led the battle against the last Borg cube that invaded Federation space.
After Worf is beamed aboard the Enterprise, he is informed by Picard that they could use some help at tactical, Worf's old position before his transfer to Deep Space Nine. No mention is made as to what happened to the previous tactical officer, as the Enterprise was not shown suffering any damage that would have resulted in injury to a member of the bridge crew. Furthermore, when the senior staff is assembled earlier, the chief of security/tactical officer is not present, almost as though the Enterprise-E did not staff this highly important position in anticipation of Worf's return.
The Enterprise disguises its warp signature with the moon's gravity when the Vulcans approach Earth. However, at the same time that the Phoenix jumped to warp (the event that caught the Vulcans' attention), the Enterprise was out in the open and even firing its weapons. If the Vulcan sensors could pick up the Phoenix's warp signature, surely they could detect a Sovereign-class ship's warp signature and a volley of quantum torpedoes, not to mention the transporters used to get the Earth-bound Enterprise crewmen back to their ship.
In the last scene aboard the bridge, Picard orders Data to lay in a course for home, which he does. Data is seated at the operations station, not the helm. There is conspicuously no helmsman on duty at the time. Even with the death of Lt. Hawk earlier, there should still be plenty of qualified pilots who could take that station.
Zefram Cochrane puts his hand on the body of the telescope when Geordi invites him to look at the Enterprise. No one who knew anything about astronomy would ever do this as it would change the alignment of the telescope.
When Picard walks around the ship with Lily and she asks how big the ship is, Picard says it has 24 decks. But about 6 minutes earlier in the movie, Lieutenant Daniels tells Worf that the Borg have control over "decks 26 up to 11". The film's text commentary suggests several possible explanations for this inconsistency, including the idea that decks 25 and 26 were top secret, so Picard had to mislead Lily in case the Borg assimilates her. It's also possible that decks aren't numbered in a sequence, so that a ship with 24 decks can actually have deck 26. However, no convincing explanation exists.
After the destruction of the Borg sphere, Picard asks where the Borg were firing on, and Riker heads towards the Science station on his right to read the coordinates from the display. Just a moment after, he asks Lt. Hawk (who is at the helm) for the damage done on the surface. Hawk states that "long-range sensors are still off-line", but at that distance away from Earth, short-range sensors would have sufficed. Furthermore, Hawk shouldn't even be capable of reading long-range science sensors from the helm console, and even if he could, Riker is still next to the science station, with the science officer next to him, AND the information about the surface is still on the display, so there is no need to ask the helmsman for it.
Picard misquotes Moby Dick when he says, "And he piled upon the whale's white hump the sum of all the rage and hate felt by his whole race. If his chest had been a cannon, he would have shot his heart upon it." The actual quote by Herman Melville reads, "He piled upon the whale's white hump the sum of all the general rage and hate felt by his whole race from Adam down; and then, as if his chest had been a mortar, he burst his hot heart's shell upon it."
When Picard refuses to order self-destruct, insults Worf and then storms off the bridge, Dr. Crusher simply accepts his irrational behavior and tells Lilly the discussion is over. This is seriously incongruous with her well-established character. She would have pursued Picard and berated him as Lily did.