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 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.
When the Borg Queen asks Data how long it's been since he last experienced forms of pleasure, he replies that it had been "8 years, 7 months, 16 days, 4 minutes, 22 seconds" (referring to when Tasha Yar takes Data to bed in Star Trek: The Next Generation: The Naked Now (1987)). "Naked Now" takes place on Stardate 41209.2 (which translates to March 16, 2364 in current calendar format), while this movie takes place on Stardate 50893.5 (November 22, 2373), which would actually make it 9 years, 8 months, 6 days.
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".
On the Defiant when Worf yells "Report!" he's on the ground just holding the Captain's chair arm, then the camera changes and he's suddenly in the chair. Next, when Worf is saying "..day to die!" some one is behind Worf moving toward the bridge door to leave, but disappears when the camera switches to the angle with the helmsman, but the person appears again when Worf says "Prepare for ramming speed!" and finally leaves the bridge.
When Picard goes into the conference room to work on new modulations he and Lily enter it from the left side of the Bridge. When they go back onto the Bridge after Lily convinces him to blow up the ship, they walk through the same door in the conference room, yet come out on the right side.
When Picard tries to convince Lily that he is a friend and asks for the phaser, she is seen starting to lower it and hand it over to him. The next time the camera is on her, she is clearly pointing the phaser at him. Only abit later does she lower it and hand it to Picard.
Early in the film, when Deanna finds Dr. Cochrane, it is still night. Immediately after she passes out, the very next shot is one from space of North America in daylight. (While it is arguably the case that the stories on land and on the Enterprise diverge from temporal sequence, it is too early in the film at that point for them to have done so.)
In the deflector dish scene, Picard is seen shooting a section of the dish to blast one of the Borg with a jet of plasma or gas from that section to send that Borg off the ship into deep space. Later, he's seen making a wild, non-magnetized jump across the dish to the other side without any form of control except for the magnetic boots, across the same jet of plasma that blasted the Borg off, without it affecting his course. Even with the idea that the jet of plasma would be a lot weaker, it would still throw off his aim.
When Worf confronts Picard on the bridge and declares that if "he [Picard] was any other man, he would kill him," Worf does not have his sash on. In the bridge scene where Picard decides to destroy the Enterprise Worf has his sash on. Possibly, when Worf was ordered off the bridge he went got it and put it on before returning.
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.
At first glance, there is no reason for the Borg to travel to Earth in the 24th century. If their plan was to invade Earth in 21st century, the Borg could have done their time-travel from a safe faraway place, then travel to Earth, or just send the Earth's coordinates to the 21st century Borg so the latter invade the Earth instead. However, even though there was no Federation in the 21st century, its future territory was occupied by several advanced space-bearing species, often at war with each other (Vulcans and Klingons in particular). Apparently the Borg figured that a small sphere has less chances of reaching Earth in the 21st century than a fully armed cube in the 24th. The point is that they made the right choice - the idea with the cube worked just fine and Starfleet failed to stop the sphere. It is also possible that after incorporating the time machine and weapons, there was no place left in the sphere for warp engines or long range communicator. In any case, we don't really know Borg's actual plan. The Queen said that they had many goals. Perhaps pulling the Enterprise into the past for safe assimilation was one of them.
How did Picard know about the weak point in the Borg cube? Before transmitting the coordinates for the point, Picard had a moment of "Borg voices in my head". Apparently he got the coordinates from this chatter. Perhaps the Borg were just reporting some dangerous power buildup in this spot, which just turned it into an Achilles heel, and Picard intercepted this message and acted upon it. Whatever that information was that Picard heard, either the Borg didn't know he still had access to their communication channels (so they didn't block him), or they knew, but ignored his (mental) presence, as they usually do, by the way, until being directly threatened.
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 Star Trek: The Next Generation: Time's Arrow: Part 2 (1992) as well.
While Picard, Worf and Hawke are trying to disable the "interplexing beacon" the Borg are building on top of the deflector dish, smoke or steam can be seen rising up in such a way it only can in an atmosphere. Steam/smoke would normally dissipate in a vacuum, zero gravity, in all directions, if it's not propelled, and if it is propelled "upward", then it will continue to "rise" and won't come back "down" as seen in the movie. However, all this is true if the electric charge is taken out of the equation. The steam is likely ionized (positively charged), if not for functional purposes (we know that plasma is often used in the deflector), then because of Picard's phaser fire. Picard also mentions that the dish is charged with anti-protons (negative charge), which perfectly explains why the positively charged steam is pulled back toward the dish. Plasma behavior in an unknown electromagnetic field can be complicated enough to explain all the apparent inconsistencies of its motion in space.
At one scene, Zefram Cochrane looks at the Enterprise through the telescope from Montana. Two minutes of movie time later, Picard shows the Earth to Lily Sloane, and the ship is over Australia on course to North America (Picard remarks that Montana is coming up soon). However, between those two scenes various incidents of crew fighting the Borg are shown, indicating that some time has passed by. A ship at the presented orbit can go from Montana to Australia in less than a couple of hours. Moreover, the stories on the ship and on the ground aren't connected at that point, so theoretically the second scene could have happened before the first. In any case, there is no obvious error in the ship not being over Montana in the second scene.
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 Star Trek: The Next Generation: Birthright: Part 1 (1993).
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.
During the deflector dish sequence set on the outside of the ship, both the sparks that come from a shot Borg and the dust that comes from releasing the Borg transmitter clamps fall to the deflector dish "floor".
The pylons of Cochrane's warpship do not provide a free area of space above or below the ship to generate a warp field between the warp nacelles. Instead the pylon allow the ship's body to stand between the two warp nacelles. According to the pseudoscience of Star Trek, a warp field can not be generated under such conditions (in other words, Cochrane's ship shouldn't be able to go to warp because of its design).
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 the spaceship 'Phoenix' takes off the people are shown looking at the take-off. At this distance, the people would be all vaporised by the rocket exhaust, or at least be killed by the hot and toxic gases produced by the missile.
When the Phoenix goes into warp, stars appear to be streaking past the ship. The interstellar distances are so huge that even at high warp (let alone Phoenix's low warp) the relative motion of stars would be imperceptibly small to the human eye. Of course, this "error" appears in almost every Star Trek television show or movie. An acceptable explanation is that all this streaking is just some weird optical effect created by the warp field. Since the whole concept of warp drive is well outside of the real physics, it would be pointless to speculate about the nature of this effect. As for the apparent shifting of the stars at sub-light speeds, it's perfectly explainable by the change in ship's trajectory.
When the Phoenix drops out of warp and turns around, we see the Earth and the Moon as they would appear from a distance of just a few million kilometers away (based on a fact that even the Moon's disk is still discernible). It is claimed that a ship in warp should have traversed a much greater distance. However, that's not necessarily true. Although we don't know how fast the Phoenix was going and for how long, it's safe to assume that it didn't go beyond warp 1 (which is equal to the speed of light). Also, the movie shows only 10 seconds of flight (two 5-second cuts), and there is no reason to suggest it took more time than that. This results in a total distance of 3 million kilometers, which is highly consistent with Earth's appearance. Moreover, it was generally assumed that the ship was moving away from Earth the whole time. But when we see Earth through the window, it appears to be growing, indicating that Phoenix is moving toward it. Either the warp flight was on a curved trajectory (we know from the series that it's possible), or that Riker's "throttle back" pushed the ship into reversed course. In both situations, the argument in favor of the goof is weakened even further.