As Beckett and Castle are visiting the Plastic Surgeon, it is plainly daytime through the office windows. However, Beckett then receives a phone call from Detective Ryan who is starting a raid on a private boat, but his scene is taking place at night. Then it cuts backs to Beckett in conversation still in the daytime.
When Lanie calls Beckett to tell her about the tattoo, Backett has a gray jacket. When she meets Lanie, the jacket is black.
Surgical implants have part numbers, not serial numbers.
In the episode the victims are found hanging from the neck. In the briefing later in the episode Captain Gates and Detective Esposito say the victims were "hung" however when speaking about a human being killed this way it is proper to say "hanged" not "hung. You "hung" a picture you "hanged" a person. Even though he didn't say it himself, as a writer, Castle would have known this enough to correct it.
When Lanie and Espo talk about the doubles she recalls a strange thing and says "someone at work got promoted". But they work together and Espo most likely knows that person and Lanie would have mentioned the person's name instead of "someone"