Factual error: In the Paris meteor scene, the POV is from the walkway of Notre Dame cathedral, with a very famous gargoyle in the RH side of the frame. From that POV, the Eiffel tower is visible, but the L'Hôtel national des Invalides (Napoleon's Tomb, the gold-domed building in the foreground) is in the wrong location. To show both of those structures in those positions, Notre Dame would have be located about a mile southwest of its actual location.
Revealing mistake: When Ben Affleck riddles the shuttle's fuselage with the bullets from armadillo's gatling and then rams it to get outside, it can be seen that the side wall of the ship is paper thin. Now, I know titanium is a sturdy stuff but don't believe such a flimsy construction would be enough to stand the ordeal of a space flight with gravity loads during the start, escaping/entering the atmosphere and pressure difference in the vacuum of space.
Visible crew/equipment: During the scene when the crew is trying to disable the bomb, the camera pans in from a wider shot to a close up twice, once at the start of the scene and once towards the end. During the second pan, on the left side of the screen, the matte box or some other part of the camera can be seen.
Continuity mistake: When AJ bumps into Bear his shirt is very dirty on his right side, but as he runs up the stairs it is much cleaner and has a sweat mark around the neck line. Next shot on the tower and the sweat is not in the same pattern. After he slides down the cable the dirt is in a different pattern again. (00:14:10 - 00:15:00)
Continuity mistake: When the team meets the flight crews for the shuttles, there are two munitions specialists - Gruber and another astronaut. We see this other astronaut board the shuttle, then he's never seen again, not even his body.
Character mistake: During the scene when AJ is yelling out for any other survivors in his crashed shuttle, we hear Lev calling "A.J.", but with no Russian accent. Then Lev gets his accent back when he says "I'm here." (01:28:10)
Deliberate mistake: When the bomb is going to be remote detonated, on the inside lid of the briefcase, the remote destruct warning says that upon activation, the warhead will detonate in T minus 10 minutes. When the countdown begins on the bomb however, it is only for 6 minutes.
Audio problem: When the Russian space station is on fire, and the crews are running out, you can hear Harry say, "Where's the rest of the team?" but his lips don't move.
Revealing mistake: When AJ drives through the wall in the spaceship wreckage, the material flaps about in a very un-metallic way.
Continuity mistake: When Grace throws down Thurman because of frustration, she first grabs him by the back, but in the next shot he's facing upwards. (01:55:25)
Factual error: The military sends a signal to the drilling team's nuke, arming it and triggering a countdown. A sudden rush to disarm the warhead ensues, with the NASA team on Earth temporarily breaking the uplink and stopping the detonation countdown. However, modern nuclear weapons do not require a "constant uplink" to detonate. Once the signal is sent and the nuke is armed, the signal can be cut or interrupted and the nuke will continue counting down.