Character mistake: In the opening scene is a paragraph describing why the Top Gun school was started. The word "insure" is used when the word should have been "ensure". They wanted to guarantee the U.S. had superior pilots, not take out an insurance policy.
Factual error: In all sequences where Maverick "puts on the brakes", he is shown pushing the throttle quadrant forward and pulling the stick back. This would put him in a full-afterburner climb, but you don't really stop and it doesn't work in a medium speed tail chase (which is easier to film). The air brake does effectively nothing on the F-14 because it is too small, and not deployable at full power.
Continuity mistake: Take notice of everyone's hands during the action sequences. No one is wearing gloves. As standard issue for military pilots, the gloves are made of Nomex and provide protection from fire and allow for a better grip when things get a little hot and heavy.
Factual error: Pretty much everyone who fires a missile in this movie uses the stick's trigger. In a fighter aircraft, the trigger controls the radio/intercom or fires the gun. The red button on the far left fires weapons after the hat or castle switch selects them.