At the very beginning, they show a castle during the credits, then "In Edinburgh In 1831-" then show a closer up of the same castle and a horse and carriage, and you can clearly see two or three automobiles parked next to the castle.
The scene in the pub right after it's announced that MacFarlane sold Gray's horse and buggy for a mere four pounds and the new owner is embarrassed into buying him a drink. Right after the barmaid goes to fetch him one, Fettes enters the pub to talk to the Doctor. When the barmaid returns, she has two drinks. Yet she had no indication that MacFarlane was meeting anyone, and the doctor didn't know Fettes was looking for him.
When Gray kills Joseph, the latter's right arm is stretched out from his shoulder at a 90 degree angle and Gray reaches over to shoo away his cat who begins to nuzzle Joseph's right hand. In the next shot, the right arm of the dead Joseph has suddenly moved to a position where it lies parallel to his body.
When MacFarlane and Gray gaze into the mirror and we then see their reflections in a separate shot, the partings in their hair change etc as one would expect on seeing their reflections, but Gray is now sitting on the opposite side of MacFarlane to where his reflection would actually be - i.e. Gray is sitting on MacFarlane's left so would be reflected as sitting on the right of MacFarlane reflection, not reflected still sitting to the left of MacFarlane's reflection like in the movie. The reflections were obviously filmed after a cut and the actor's swapped places in error to film that scene.