In the initial confrontation with the Borg, Riker orders, "Full reverse!" This order is never countermanded. Yet when the phasers disable the tractor beam, the Enterprise remains stationary instead of immediately moving backwards.
A Borg drone beams in near the emergency turbo-lift, attacking Riker and Worf. When he beams out, he's near the Ready Room.
When Riker is contacting Admiral Hanson on the desktop computer, the close-up image of the computer is recycled from stock footage taken for season 1 (with new footage of the admiral superimposed on the screen), while the wide shots show the prop, which had been altered since the stock footage was taken. A series of gray stripes had been added to the prop. The stripes then appear and disappear between shots as new footage is inter-cut with the old.
When the Borg cube moved to intercept the Enterprise, it traveled with an edge pointing forward. (Mostly likely this was for better, more dramatic visuals.) However, in Q Who (1989), what was presumably the same cube moved with a flat side forward. Then for the rest of "The Best of Both Worlds", the cube moved with a flat side forward.
When Riker and Shelby are in the turbolift, Riker orders it to halt en route to the Battle Bridge. Shelby orders it to resume, after she 'threatens' Riker, and it does so. Riker outranks Shelby so the turbolift should not have obeyed Shelby's order. However, the turbolift is not given a specific command authorization, instead just a simple verbal command. It can be easily be inferred that, under ordinary circumstances, the turbolift does not give any specific weight to the rank of the person giving commands. It would be akin to a modern elevator: anyone can push the "door closed" button, and anyone can then push the "door open" button, regardless of positional authority.
When the Borg are intercepting the Enterprise, long range visuals shows one face of the cube fairly well lit, and the adjacent face dark. (Of course this was for better, more dramatic visuals.) In reality, unless the Borg happened to be fairly close to a star, there would have been no light source to make any side better lit than the others.
When the Borg transport off of the bridge, they all have a shimmering green light form over them. The dead Borg simply fades away. It's depicted in Q Who that dead Borg don't transport back, they disintegrate. It even leaves a bit of a stain.
Worf states that the Borg's course will lead them to Sector 001, "the Terran system." Systems are designated by their stars. He should have said the Sol system, not Terran. However, this is probably common usage. Humans (and those raised by humans, such as Worf) will name other systems according to stars seen from Earth but will name their own system according their home, i.e., Earth or Terra. Moreover, 'Sol' isn't really a name: it merely Latin for 'sun'.
When Shelby is showing the Enterprise staff footage from the Borg attack in the briefing room, the image can be seen reflected on the table. When footage of Admiral Hanson plays on the same screen, no reflection is seen. This is due to the footage of Hanson being superimposed in post-production, while the footage of the Borg attack was played live on-set. This has been corrected in the HD remaster via a reflection being added to the table to match the footage of the admiral.
(At 16:40) In the Ready Room adjacent to the Bridge and with urgency, Picard orders Commander Data to jump to warp 9. A full minute later and until the end of the scene, the stars outside the window remain static.
The Enterprise crew first learns that New Providence had been destroyed. But why have the ship's scanners not realized it much earlier?
During the poker game, all cards have been dealt. Wesley is showing 3 of a kind and bets 10. Shelby only has 2 pair and just calls. This doesn't make sense as she is already beaten by Wesley's 3 of a kind. She should have either folded or raised to represent she could beat 3 of a kind. Just flat calling was not a suitable option.
The Enterprise hides in a nebula, which Data says will provide screening against the Borg's sensors. A few moments later, Worf announces that the Borg are continuing their scans. The Borg's sensors are more advanced than the Enterprise's. If the Enterprise can collect sensor data on the Borg's activities from within the nebula, the Borg should be more than capable of detecting the Enterprise.
Riker encounters Shelby after he went directly to Picard about separating the saucer section. The pair then enter the turbolift near the Ready Room to proceed to the Battle Bridge. However, the other turbolift, on the starboard side of the front of the Bridge is specially designated, connected directly to the Battle Bridge.
Riker attempts to lead the away team to the Borg ship and gives command of the Enterprise to Lt. Cmdr. Shelby. When Troi points out that Riker is now captain and it's inappropriate for him to lead the away team, Riker elects to remain on the bridge and puts Shelby in command of the away team. Shelby is not part of the Enterprise's command structure and does not outrank Data, the ship's second officer (and acting first officer due to Picard's abduction). It is not appropriate for her to be given command over him.
When Admiral Hanson turns down Riker's offer to play poker in his quarters, look at the shadow underneath Riker's left hand. When Hanson says "Rumor has it Commander Shelby has played a hand or two" the shadow of the microphone's tip is visible. At first it appears to be the shadow of Riker's hand, but closer examination shows that this shadow moves independently of Riker's.
O'Brien tells Riker that the coordinates he's at correspond to the center of town. But the next shot shows no buildings, only a giant pit, meaning that the Borg scooped up the entire town. But if Riker is supposed to be in the center of the city, then he should be at the bottom of the crater.
Picard states that the Emperor Honorius witnessed the fall of the Roman Empire and the end of Roman civilization. In fact, Honorius was Roman Emperor during the sack of Rome but was not the last Roman Emperor. The Roman Empire would persevere for another fifty years after Honorius until falling completely in 476 AD.
When the Enterprise is preparing to leave the nebula, Captain Picard orders forward torpedo bays loaded. The Enterprise is preparing to run from the Borg cube. The Borg would be behind them, not ahead. Aft torpedo bays should have been ordered loaded as well.
After the Borg abduct Captain Picard and Worf announces that they are fleeing at warp, Riker orders "Maintain pursuit." Moments earlier, the Enterprise was being pursued by the Borg ship, not the other way around. They cannot maintain a course of action they have not yet embarked on. He should have ordered a pursuit course and engaged.