These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Harry Stamper: And this is the best that you c - that the-The Government, the U.S. government can come up with? I mean, you-you're NASA for cryin' out loud, you put a man on the moon, you're geniuses! You-you're the guys that think this shit up! I'm sure you got a team of men sitting around somewhere right now just thinking shit up and somebody backing them up! You're telling me you don't have a backup plan, that these eight boy scouts right here, that is the world's hope, that's what you're telling me?
Crowning Music of Awesome: Lots of people complained about the science goofs and the plot holes. Film critics were absolutely merciless and wrote scathing reviews. But the Trevor Rabin soundtrack may have been worth the price of admission all by itself.
"Funny Aneurysm" Moment: In the first ten minutes of the film, New York is decimated by a meteorite shower (See Harsher in Hindsight). It's kind of worth mentioning that in the middle of this scene there's a cabbie that screams something with each impact: "Look at that! Whoa! We're at war! Saddam Hussein's bombing us?" Oh, the innocent irony of 90s catastrophe films.
Made even more ironic by the fact that, despite Saddam Hussein's lack of involvement in 9/11, America still ended up going back to war with him less than a decade later...
Harsher in Hindsight: very disturbing to see the World Trade Center twin towers with huge gaping holes in them after the events of 9/11.
Not to mention the destruction of the Space Shuttles Atlantis and Independence after the Columbia disaster.
In the beginning of the movie, Truman mentions that NASA does not have the power to detect all incoming objects. On February 15, 2013 a previously undetected asteroid 20 meters in diameter and weighing over 10,000 tonnes exploded above Russia in the largest airburst since the Tunguska event of 1908. Furthermore videos of the airburst looked very similar to scenes of meteors falling in Armageddon.
Nightmare Fuel: those are people falling out of the Chrysler Building during the meteor shower in New York City. For additional squick, watch the movie frame per frame. That thing that just fell on a cab's trunk, severely deforming it? Not a piece of building.
Tear Jerker: for a testosterone-fueled Action Movie, there are several, but the tearjerking scene mostly involved Grace and Harry.
Special mention for the music video for the song I Don't Want to Miss a Thing, where they mirror the most tear jerking scene in the entire movie, only with Liv and Steve Tyler...
Chick's reunion with his son is another.
FOR ALL MANKIND
"That salesman is your daddy."
Naming off the casualty list for the mission alone is enough to bring tears.