Character mistake: When everyone is preparing to go in to dinner, Robert is wearing an adjustable bow tie. It's bad enough he's wearing black tie to dinner when he has guests, as his mother points out, but Robert, the Earl of Grantham, absolutely would not be seen wearing an adjustable tie with a stand up shirt collar, even among only his immediate family.
Factual error: When Lady Edith visits a Harley Street doctor his nameplate is shown outside his surgery. His name is prefixed with "Dr." However, one of his post-nominal qualifications is "FRCS." As a Fellow of the Royal College of Surgeons, his name should be prefixed with "Mr.", as with all qualified surgeons in the British Isles.
Factual error: There's more PIR's (movement detectors used in security systems) in Hireclere than you could find in a security warehouse! Just look behind the actors in the corners of a room about 8 feet up. White plastic things about the size of a packet of smokes. I think the earliest units on the market started appearing around the early 70's. The PIR's in Downton Abby look to be around the late 80's onward - pretty advanced for 1914.
Continuity mistake: When Cora (Lady Grantham) goes to Violet (the Dowager Countess)'s house to discuss Matthew and Isobel Crawley, she sits at the table and takes off her gloves shown in a shot from the front, and then the shot changes to over her shoulder and she is seen taking the same glove off again.
Continuity mistake: Near the end of the episode with the garden party, there is a scene where Edith is pleading with Anthony Strallan not to leave. You can see Lady Grantham lying down on a lounge chair in the background, with a man who is distinctively Lord Grantham standing next to her talking to her. But the scene immediately cuts to Lady Grantham, and her maid O'Brien walks over and Robert is nowhere to be seen.