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.
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.
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.
President Whitman's statement of "Today we celebrate our Independence Day!" was ad-libbed by the actor.
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 Smith. (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)
What Could Have Been: 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 as opposed to a suicide mission that just happened to be plot significant.
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...