Continuity mistake: When Bart falls down the stairs, you see him fall down at least 50 stairs in the close up. There are never this many on the full shots of the staircase.
Continuity mistake: When Lisa arrives at Dr. Nick's, amongst the wheelbarrow line is a Professor Frink (and he's not even first in the cue). However, just a few minutes later, when Lisa ends up in the mud pit, we can see Frink amongst the crowd.
Answer: Willy referred to an encounter he believed he had with actual aliens, while Wiggum thought he was referring to the arcade game.