When rushing to get in the ship the first time, Wallace kicks away the paint-stained ladder, which falls to the ground as they take off. Yet when they land on the moon, they use the same paint-stained ladder to get in and out of the ship.
The packaging of the Crackers packets change color between being moved from the kitchen into the rocket. When upstairs, the packets are orange and the "Crackers" writing has white text against a black background. Before take-off, it is black writing against a gray background.
Wallace paints the same part of the rocket twice.
While Wallace and Gromit are traveling in a rocket ship in space, they are seen walking around. They wouldn't be able to breath unless they have an oxygen tank supply on their rocket ship, Also they would actually be floating around in their rocket ship due to no gravity unless they have an artificial gravity machine.
If Wallace and Gromit were walking on the moon, they would be six times lighter than on the Earth and would actually bounce and float higher in the air which this film doesn't allow.
The mice that were downstairs in Wallace's house when he was starting to build the rocket, they were gray mice, but when Wallace finishes building the rocket, the mice were now pink.