Actor Allusion: If you think Jeff Goldblum's "must go faster, must go faster" line as David and Steve are escaping the alien mothership sounds familiar, it's because they used the dialogue loop from Jurassic Park when Goldblum says the same line.
Backed by the Pentagon: The producers tried to use this trope, but the Pentagon refused to back a movie whose plot involved Area 51, (because it doesn't exist!). Tellingly, Emmerich noted in one interview (and in the DVD commentary) that the Pentagon bent over backwards to help him get the technical details right until they saw Area 51 mentioned, at which point they pretty much told him that he was on his own.
The existence of Area 51, which is Groom Lake Airfield, Nellis Air Force Base, has never been secret. The fact that you can clearly see it from the perimeter fence kind of makes such a denial pointless. What they do or don't do there is another story.
A deleted scene (included in the extended version) early in the movie had David explaining exactly how our satellites were being hijacked by the aliens. At the end of the scene, Harvey Fierstein's character planted an (ad-libbed) kiss on David. Ironically, it was Roland Emmerich himself who decided to cut the scene, lest he incur the wrath of the MPAA. And the kiss in question was at best a platonic kiss, so it wasn't even that close to potentially upsetting the MPAA, especially seeing how another thing that Fierstein was in got away with a similar male-male kiss without it being censored.
A scene was cut which would have explained that we based our personal computing technology was based on what we'd reverse-engineered from the crashed alien ship, which is why our computers can interface with it. It was cut because Emmerich didn't think people would make that big a deal of it. It's probably the biggest Head Scratcher surounding the film.
Yet another deleted scene (according to an interview with Judd Hirsch) would have explained that Julius Levinson was a Rabbi who lost his faith after his wife died. This is why David is so shocked by his revelation that he hadn't, "spoken to God" since then. It's also why he takes it upon himself to lead the prayer circle in the Final Battle.
Star Fox 64's Katina stage features the Saucerer, a familiar looking flying saucer that fires a giant Wave Motion Gun if you don't beat it fast enough. You fight it in a giant aerial battle with many allied and enemy ships. The official American strategy guide even lampshades this, saying that the Saucerer's plans might have been stolen from another alien race.
Production designer Patrick Tatopoulos came up with two completely different designs for the aliens. Roland Emmerich loved both sketches and asked to use them both, leading to the distinct "small alien in an exoskeleton" look.
And Harvey Fierstein's "Oh, my God. I gotta call my brother, my housekeeper, my lawyer. Nah, forget my lawyer..." was ad-libbed, although it was original "nah, fuck my lawyer," and then dubbed over (rather obviously).
Similarly, Fierstein's character's planted kiss on Goldblum's character was ad-libbed by Fierstein. Of course, it ended up cut in the theatrical cut, but restored in the Special Edition.
The Rousing Speech Whitmore gives to the pilots before the climax was not intended as the final version of the speech. Emmerich and Devlin got caught up in filming they forgot to keep working on it up until Pullman was doing rehearsals on set. But Pullman was just so bombastic that the extras and crew were applauding, and they left the speech alone. The line "Today we celebrate our Independence Day!" was ad-libbed.
Will Smith's "AND WHAT THE HELL IS THAT SMELL?!" line while ranting about having to drag an unconscious alien through the desert in his parachute when he could have had a day off for the Fourth of July was not in the original script: the desert setting that they were filming in was actually the salt flats near Great Salt Lake in Utah. Great Salt Lake is home to brine shrimp. When they die, the bodies sink to the bottom of the lake (which isn't very deep) and decompose. When the wind kicks up just right, the bottom mud is disturbed and the smell of millions of decaying brine shrimp can be very very bad. Apparently, nobody warned Will. (Roland Emmerich also said that he would have loved to let Smith improvise as much as he could, given how hilarious the scene turned out).
Will Smith also ad-libbed his reply of "Just tell them I hit you." to the burly soldier.
The original ending involved Casse being disallowed to fly alongside the others, and so he straps an explosive to his biplane and performs the Heroic Sacrifice by flying it into the superlaser as per the ending that found its way into the film. It was determined to be a little bit too comical seeing a biplane match the speed of the fighter jets, and the reshot version allows us to see Russ make his decision for a Heroic Sacrifice as opposed to a suicide mission from the outsetnote The movie's novelization contains this alternate ending..
Homaged/Parodied in one of the later Metal Slug games as a cutscene once the Big Bad ship starts to fire its big beam. An antique plane flies into the beam and...
Kevin Spacey was Dean Devlin's first choice to play President Whitmore, as they were high school friends. Matthew Perry was also considered to play Jimmy.
There was a scene in the film that would have explained one of the biggest Headscratchers of how Levinson's Macbook was compatible with an alien operation system, by showing that Levinson analyzed the computers on board the crashed alien fighter to build the virus. For some reason, the scene was removed from the theatrical cut.