Revealing mistake: In the scene where Costello is smashing Billy's hand with the boot heel, you can tell from the position of Billy's arm that Costello is hitting the pool table rather than Billy's hand. He is actually hitting his own hand more than he could possibly be hitting Billy's. (00:36:00)
Revealing mistake: When Costello's men drive off, after throwing Queenan off the top of the building, the state troopers are firing at the van. One back window is shot out, then in the next shot of the van the other window is shot out and then the same one is shot again so the one window is shot out twice in the same scene.
Revealing mistake: ***SPOILER*** When Costello is killed, a blood-pack is noticeable under his shirt.
Revealing mistake: In the shootout at the elevator near the end of the movie, when Sullivan shoots the last guy, you can clearly see kneepads under his pants, to soften the fall.
Revealing mistake: In the Porn theater after Frank reveals the dildo to Sullivan and they argue about it, when the camera zooms out and the view is behind Sullivan the guy to the far right looks like he was added post production.
Revealing mistake: When Mr. French smashes Costigan's arm cast, on the second attempt Costigan seems to be using his own force to smash the cast. As French loses his grip at the top of the motion, Costigan's arm moves downward with all the force it would have had had French not let go.
Revealing mistake: Costigan carries a Walther PPK/S. The hammer on his pistol is down both before and after he shoots the junkie armored car thief. The shot to the leg should have automatically recocked the hammer.
Answer: The FBI and police are separate organizations that do not share full information. Only the FBI and Costello himself knew about the informing, and the FBI can only do so much to stop Costello being caught without drawing attention to their relationship. The police and Costello's gang are fighting a separate battle, hence the two moles.