Continuity mistake: They mention that the goose has hatched out 7 goslings. They say there were 8 eggs and one didn't hatch. The one egg is given to Templeton, and it is later smashed open. Later that night as they're going to sleep, the mother goose has 7 goslings under her, but then then it shows Jeffrey (one of the goslings) sleeping with Wilbur. Seems they had an extra gosling with the mother.
Character mistake: Near the end of the movie, Charlotte says she doesn't have the strength to climb down into the crate. Two seconds later, she climbs down to comfort Wilbur, then climbs back up. Even if she couldn't go all the way to the crate, she was low enough that Wilbur could have carried her there himself.
Deliberate mistake: Throughout the movie we hear every animal (and insects/arachnids as is the case with Charlotte and her offspring) with one exception. Every time Charlotte catches her meals in her web we never hear the fly or other insect she is descending upon cry for help, or plead with her to be spared. This is deliberate due to this being a kid's film but still bears noticing as a mistake due to the fact that they are the only ones rendered mute for plot convenience.
Other mistake: Fern's surname is misspelled as "Arabel" in the DVD subtitles. Her surname is actually spelled "Arable."