Revealing mistake: The two ships that the astronauts and drillers use are supposed to be new "X-71" military vessels that were top secret - and just happen to bear a resemblance to the actual NASA shuttle orbiter. But, when they are launched, it's modified shuttle footage, and in at least one of the long shots they didn't add the X-71 overlay. The key distinction is the two top-mounted booster rockets on the shuttle that the real shuttle doesn't have. There are also a few shots of the firing engine on the mission control screen that is the unmodified shuttle and not the X-71 configuration or paint scheme. This occurs later as well during the MIR docking sequence when mission control shows footage of a real docking arm that looks nothing like the soft-docking arms of the station.
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.
Revealing mistake: When AJ drives through the wall in the spaceship wreckage, the material flaps about in a very un-metallic way.