Revealing mistake: In some extended edition cuts of the film, Regan backwards spider walks down a flight of stairs, and blood falls from her mouth. But her face isn't that of Linda Blair, but of a stunt double.
Revealing mistake: When Regan meets Father Karras, she projectile vomits in his face. But if you look closely while this vomit happens, you can very easily notice that it is the stunt double.
Revealing mistake: Crash mats are visible on the steps, when the Father is thrown through the window and rolls down the stairs.
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.
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)
Revealing mistake: When Father Karras sprinkles holy water on Regan, they forgot to paint her ear lobes and it looks terrible. (01:32:30)
Revealing mistake: As Chris walks around the attic to see what's making the noise the shadow of the candlestick appears on her face which is not possible as it's the only light source up there. (00:30:15)
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.
Answer: The demon wants everyone to lose hope for the girl and give up on her. Her father, having already flown the coop, just makes this goal that much easier.
TonyPH