In the White Princess, Richard Pole has a useless, crippled right arm. In the Spanish Princess, he has full use of his arm.
There are many historical errors in the series, primarily the fact that Henry VIII was only 10 years old when his brother died.
Throughout the series, Thomas Boleyn is referred to as "Wiltshire," the name of his Earldom, likely to avoid spoiling that he was indeed the father of Mary and Anne Boleyn. However, the series picks up in 1501 and Thomas Boleyn was not granted the Earldom of Wiltshire until 1529.
There is a 26 year age difference between Harriet Walter and Eliot Cowen who play Margaret Beaufort and Henry VII. In real life Margaret had Henry at 13.
Throughout the series, primarily in Season Two, Henry VIII refers to Margaret Pole as "Lady Aunt" or "Aunt Maggie" despite the fact that she was his first cousin, once removed as she was his mother's cousin (the women having been children of the brothers Edward IV and George Plantagenet, Duke of Clarence.)