Throughout the movie, characters speak in Aramaic. However, this particular form of Aramaic is the Syrian variant, spoken in that region in during the VII and VIII centuries and not the I century Palestinian form, which is what people in the times of Jesus would have spoken.
As always, differences with the written source material, especially relating to artistic decisions, are not counted as goofs. Historical inaccuracies are similarly exempt, especially when caused by reliance on sectarian traditions or Renaissance/Baroque art.
Jesus' arm is dislocated in order to make his hand reach the nail hole, meaning that his arms should be fully extended and straight at all times. However on almost all shots of him on the cross, his body sags forward with his arms bent at an almost 90 degree angle.
Satan is seen moving through the crowd while Jesus is being beaten. Jesus is the only one who is supposed to be able to see Satan, however, one man in the crowd follows Satan with his eyes as Satan moves past him.
No matter what Judas does, he cannot hurt or stop Satan's demon children who begin to harass him. Yet, as Judas is being chased by the children along a sandy stretch on a small hill, Judas scoops up a fistful of dirt and throws it back at the demons. One of the child actors is hit head on by this spray of dirt and is almost stopped dead in his tracks.
Jesus' computer-generated brown eyes sometimes fade into Jim Caviezel's blue ones. This happens when Jesus is taken at Gethsemane, and again after Jesus has first taken the cross on his shoulders (before the shot with the donkeys and palm leaves in the background).