Revealing mistake: You can see wires in the scene where Regan levitates above her bed in the exorcism scene, lifting her up. The way they pull on her dress is also visible.
Visible crew/equipment: Near the beginning of the movie, the old priest is in Northern Iraq and you can see the shadow of a microphone on his hat.
Continuity mistake: The Psychiatrist that hypnotizes Regan is wearing a tie that in the close-ups, is retied differently as the scene progresses. (01:02:05)
Revealing mistake: Crash mats are visible on the steps, when the Father is thrown through the window and rolls down the stairs.
Continuity mistake: When Chris is taking a walk with the priest Damien, she throws her cigarette to the ground, but in the next shot it's back in her hand again. (01:21:30 - 01:22:06)
Revealing mistake: As Father Merrin speaks to another archaeologist in the beginning of the movie the clock on the wall stops between ticks and he turns around to look at it. It's easy to see that the clock pendulum has been pushed up against the window from behind in order to make it stop. (00:06:45)
Continuity mistake: Father Merrin says, "I cast you out", and the green vomit that was around her mouth has now disappeared. (01:49:05)
Continuity mistake: As Regan spins her head around her blue nightgown is now totally clean. (01:49:50)
Continuity mistake: At the movie set, Chris yells, "Hold it, hold it" as Father Karras watches and walks away. In the next shot several extras along with the line of shade behind him disappear. (00:14:50)
Visible crew/equipment: In his office as Chris talks to Doctor Klein the boom mike's reflection can be seen moving around in the metal light behind him. (00:34:45)
Continuity mistake: Regan sits up in bed as Father Merrin says, "By this sign of the Holy Cross" and she has no frozen vapor coming out of her mouth. In the next shot before her head spins around frozen vapor appears and can be seen all the way down to her blanket. (01:49:35)
Continuity mistake: Father Dyer runs to Chris' car at the end of the movie as we see another car coming at them and passing on their left. In the next shot that car has disappeared. (02:05:25)
Deliberate mistake: Father Merrin says, "It is He who commands you", and we hear and see the ceiling crack above his head. In the next shot we see the ceiling above him and they are no cracks in it, but four shots later we see the exact same spot and now cracks appear even though we hear no more cracking over-dubs after the first one. (01:49:15)
Continuity mistake: When Father Karras visits his mother, he places his collar edge-wise on the shelf. In the next shot the collar is lying down at a different angle.
Continuity mistake: After Regan's head spins around, her hairstyle around the forehead changes between one frame and another.
Continuity mistake: The long curtain on the window behind her has been noticeably moved between shots as a doctor puts Regan in a trance. (01:01:50)
Revealing mistake: As Father Merrin puts his hand on Regan’s head, nasty green vomit starts to come out of her mouth. If you look closely when she turns her head towards him you can see the hose under her ear that proceeds under a false cheek and into her mouth.
Revealing mistake: When Father Karras sprinkles holy water on Regan, they forgot to paint her ear lobes and it looks terrible. (01:32:30)
Continuity mistake: Father Karras is out for an early morning jog and jogs past the same place twice as he jumps three-quarters of the track between shots. (01:02:50)
Continuity mistake: As the first part of the exorcism comes to a close, Father Karras picks the blanket up off the floor and covers Regan with it. Father Merrin's red prayer book can be seen on the left night table but in the next shot it's moved and we see his crucifix there instead. (01:53:00)
Answer: It was indeed Pazuzu, he wanted Regan to believe he was some friendly entity to gain her trust. Thus the Captain Howdy facade.