When CJ and Josh talk about pranks conducted by the departing administration, Josh remarks that on his first day, he needed a crowbar to open the drawer on his desk. However, in Debate Camp (2002), flashbacks show that Josh does not have a desk at all on his first day.
When the President meets the President-Elect, the President's lap-top computer appears, then suddenly disappears.
Otto has a full donut in his hand while talking to Josh. Then, before Josh takes a bite, the donut is only partial.
As Josh and Donna start to kiss, her purse is on her shoulder. The shot reverses and suddenly her purse is on the floor as she removes her coat. The shot reverses again and her purse is back on her shoulder.
When Josh is interviewing Lou, there is a can of Red Bull on his desk in one shot. In the reverse shot, it's a Coke can.
When Josh asks Sam to quit private practice and serve as deputy chief of staff, among the reasons Sam cites for his reluctance to return to the White House is that his fiancée, also an attorney, would have to retake the bar exam to practice law in D.C. The D.C. Bar admits already-practicing attorneys from other states without requiring them to take a new bar exam, provided that they graduated from a law school accredited by the American Bar Association and that when they took their original bar exam, they received a high enough score on the Multistate Bar Examination and multi-state legal ethics exam (multiple-choice tests used by many states as part of their bar exams). DC's admission requirements are common knowledge among lawyers on the East Coast, and both Sam and Josh should have known better.
Just before the scene where Matt & Helen Santos discuss their transition housing in their hotel room, the screen shot is of the outside of the Joseph Smith Memorial Building, obviously setting this as the Santos' hotel, although the building is actually in Salt Lake City, UT, and not Washington D.C.
After Debbie tells Santos that her sister finds him attractive, Bartlet retorts "she doesn't have a sister". In a previous episode, Debbie mentioned having a sister who died of cervical cancer. So it's obvious that Bartlet only says this to make the conversation awkward for Debbie, implying that she's the one who finds Santos attractive.