Continuity: 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 strings attached to their backs.

Deliberate "mistake": 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.

Continuity: When Homer is walking to Snake Root, he has a rolling pin in his box. You can see the pin change positions in the box.

Other: When Homer is in the Principal's office dropping out of school, the words "Principal" and "Turner" on the glass door are printed backwards (as they should be) but "Jim" isn't.

Continuity: In the scene where 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.

Continuity: When they burn down the shack where they lit off rockets, O'Dell is seen to be the last one turning around and in the next scene he turns around again.

Continuity: In the scene where the miners decide to strike, we hear the sounds of a major storm. As they walk out the door, the rain is coming down heavily. However, in the very next shot, a long shot of the same scene, there is no rain.