Revealing mistake: When Homer placed the rocket on his fence and lit it his friend asked "Should we get behind somethin'?" That's when the rocket explodes. When you see them fly back you don't need to slow motion it because you can clearly see the wires attached to their backs.
Continuity mistake: When Homer has just been given the gold medal at the National Science Fair, and he turns around to try to spot Werner Von Braun, several of the people in the crowd jump a lot closer to him when he turns back around.
Trivia: Homer Hickam's father's name is actually Homer Hickam Sr.; in the movie, the father's name was changed to John. The assumption would be that the name was changed in the movie to avoid confusion between father and son.