Continuity mistake: Halfway into the film, Shakespeare is holding Viola's face placing his hands around her jaw, looking closely at her. When the angle is behind Viola, we see that her hair gets caught around his left hand's fingers. The angle changes to a wider side angle and his hand has no hair around, even though he hasn't moved it at all.
Factual error: About 20 minutes in when they are at some ball/dance thing, Shakespeare's talking with a musician holding a lute. You can see fret markers on the fretboard of his lute, but these were not used on instruments until the late 1800's, early 1900's, definitely not in Shakespeare's time.
Other mistake: When Ben Affleck is helping the play actors rehearse a dance and Gwyneth Paltrow starts dancing the woman's part (even though she's supposed to be a man), Ben Affleck yells, "Are you a lady, MISTER Kent?" instead of "Master Kent," as men are all referred to throughout the film.
Continuity mistake: In the tavern scene, when Shakespeare and Marlowe are speaking, there is a bartender. In the shot in which Marlowe says, "His best friend is killed in a duel by Ethel's brother or something. His name is Mercutio", the bartender starts looking at Marlowe, and by the end, he turns his eyes towards Shakespeare. In the next shot, when Shakespeare says, "Mercutio... good name", the bartender's face has immediately turned toward Marlowe, without any time to do so.