When Wilbur is being forced into the crate, the straw in the crate disappears and reappears between shots.
When arriving at the fair, the lettering on Wilbur's crate has quotation marks around the phrase "Zuckerman's Famous Pig." These quotation marks were not present when the crate was shown earlier and they have disappeared again by the time the crate is in the pen.
In the beginning of the movie when Fern first gets Wilbur, and the montage of scenes takes place, Wilbur's size clearly changes from very small to almost half Fern's size in between shots. When she gets to Henry's house, Wilbur is back to being very small.
When Fern's brother Avery sits down for pancakes, his plate is empty. In the next shot (when his frog pops out of his jacket for a second) his plate has a stack of pancakes complete with butter and syrup. In the shot after that his plate is empty again and he is reaching for some pancakes.
Shortly after Avery's frog leaps over the pancakes, one of them slips out from under him. When the pancake lands on Wilbur's head, there is butter on it. In the very next shot, however, the butter has changed to syrup.
Charlotte's anatomy is incorrect; she has three body segments and antennae. Spiders only have two body segments and no antennae.
In Henry Fussy's first scene (when he is playing his violin), he wears glasses, but in his next scene (when he visits the County Fair) he does not wear glasses any more. He may have chosen to not wear glasses in the latter scene to try and impress Fern (even though this is not mentioned in dialogue during the film).
When Fern gives Wilbur his first bath, the tub is full of bubbles. When she rinses him off, all the bubbles have disappeared from the tub. In the following shot, the tub is full of bubbles again.