When Detective Beckett approaches Castle at the party and takes him to the police station, he has a closer shave than in the following interrogation scene. After returning home from the interrogation, his facial hair has returned to the much shorter look.
Beckett opens a copy of Castle's book, A Skull at Springtime, in the obsessed fan's apartment. She opens to the title page where there are lots of scribbles, but the copyright information on the opposite page is for the book Storm Fall, Castle's most recent book (also seen on the shelf in the background of the shot).
When Castle is arrested in the library he is wearing a blazer. When he is brought into the police station in cuffs, the blazer is gone.
When Castle is talking to Alexis in the limo, just before she tells him that he'll need to raise her allowance if he keeps needing to get bailed out, she pushes a lock of her hair behind her right ear, but in the very next shot, her hair is back where it was before she pushed it back.
When Castle is being held hostage by Harrison Tisdale, he is holding on to one of the cuffs of the handcuffs and the other cuff is attached to his wrist. After Castle knocks out Tisdale, he gives Beckett the cuffs when she asks him for them. He didn't have time to take off the other cuff between being kidnapped by Tisdale and Beckett asking him for the cuffs and we don't hear him unlock the other cuff in the alley.
The patches the officers wear on the sleeve shoulders say Police Dept New York City. It should say City of New York (as does the shield in the elevator).
When Castle is being held hostage by Harrison Tisdale, at one point Harrison asks Castle, "Who are you?", but his lips say, "What are you?"
The show opens with Castle at a gala book signing for his newly published book, "Storm Fall." There are piles of books, fans have bought them and are having Castle to autograph them. When Castle presents the autographed book to Beckett he tells her that he got her an advance copy. Clearly it cannot be an advance copy if it has already been published as in on sale.
Beckett opens a copy of Castle's book, A Skull at Springtime, in the obsessed fan's apartment. But she is not wearing gloves and is therefore contaminating potential evidence.