Inverted - At one point, Lily calls Picard "Captain Ahab" in reference to his personal vendetta against the Borg. Patrick Stewart was so pleased with this scene that he went ahead and played Ahab in a made-for-TV movie version of Moby-Dick.
Beam Me Up, Scotty!: Picard actually misquotes Moby-Dick. The change is obviously to simplify the text for modern audiences by putting it into more modern language. There are three possibilities: in the future, someone has revised one of the greatest novels of all time, Picard's exhausted and either misremembering or paraphrasing the quote, or Picard's translator gives it in modern English for Lily rather than the original language.
Picard: “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.”
Original: “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.”
Ira Steven Behr insisted that the "tough little ship" exchange be added to the opening battle, so it was clear that the Defiant had survived and could still be used on Deep Space 9.
When discussing what the next TNG movie should be about writers Brannon Braga and Ronald D. Moore wanted to do a movie featuring the Borg while Producer Rick Berman wanted to do a time travel movie. Their solution? Do both!
The Other Darrin: James Cromwell replaces Glenn Corbett as Zefram Cochrane, who first appeared in the original series episode "Metamorphosis". However it is now almost a given that more people know Cromwell's performance over Corbett's, making Corbett The Pete Best.
Real-Life Relative: While Jerry Goldsmith composed most of the score, his son Joel wrote some of his own cues.
Real Life Writes the Plot: The Defiant is heavily damaged and her bridge is darkened and filled with debris because the set was intended to only be shown on television and wouldn't have looked good fully lit on a cinema screen.
Recycled Set: Aside from exceptions like the Bridge and Engineering, most of the Enterprise-E's sets are actually modified versions of Star Trek: Voyager's sets. It's particularly obvious in Sickbay, whose layout is almost unchanged. (You could argue that's Starfleet design or even budget issues- why make sickbays look different on their larger ships?)
A scene with Sisko ordering Worf to command the Defiant in the battle was to be featured. It was filmed, but cut.
Picard and Riker's roles were to be switched, and more emphasis was going to be placed on the Earth storyline. Part of the reasons for the switch were apparently because Patrick Stewart felt Picard should have more of an active role in defending the Enterprise, while Jonathan Frakes wanted to have more time to focus on directing. Picard was also the one who had been assimilated by the Borg, so it made more sense for him to be the one to confront them. According to Braga and Moore's commentary, the Enterprise plot was mostly unchanged but the Earth-based story was completely altered. Also Lily (who was called Ruby in the early script) would have stayed on Earth with Picard in the original story.
The earlier drafts of the script were also quite different tonally, and more of a time-travel comedy in the same vein as Star Trek IV: The Voyage Home. When Picard and Riker's roles were swapped, the main storyline became much Darker and Edgier, with the Cochrane scenes instead used to provide moments of relief.
The Borg Queen wasn't in the early drafts of the script. They started to formulate the character when executivescript notes remarked that they were just cyber-zombies and the movie needed a more tangible villain.
The Defiant was outright destroyed in one draft of the script. Unsurprisingly, the DS9 staff objected to what effects this would create for their show, so it was changed. (Incidentally, the ship would be destroyed on DS9, but not for another 2 and a half years, and would be replaced by a look-alike with the same name.)
Tom Hanks as Zefram Cochrane. "Life is like a box of Gagh..."
In the original screenplay, Picard tells Lily her phaser was set to minimum, and would have just given him a bad rash. This original line is still found in the Novelization.
The Enterprise-E was originally planned to be another Galaxy-class—they even repainted the four-foot model with the letters NCC-1701-E—but decided later to go with the more aggressive-looking Sovereign-class.