Between 19 and 21 minutes when Molesley is introducing the class to the history subject for the term, the writing details on the chalkboard change a few times as the camera switches back and forth.
When Mary and Henry get married, the priest wraps the cloth around their hands twice. Yet when we immediately see the camera angle from behind the priest, he only has the cloth draped over their hands, not double wrapped.
When Bertie and Edith are talking he referred to her as the daughter of the fifth earl of Grantham. However Robert is the Seventh Earl Grantham.