Deliberate mistake: It's not particularly believable that the away team to stop the assassination would consist of the ship's entire senior staff, and raises the question of who was left in charge of the bridge (the answer: apparently nobody, as they return to a completely deserted bridge). Of course this is dramatic license, and at this point the audience won't care or likely even notice.
Deliberate mistake: When Cliff crashes into the pond after his first flight, Peevy tells him that he has to look around with his eyes and steer by moving his head. However, just before this as he is flying next to the passenger airplane, he looks over at it, turning his head to the left without changing directions at all.