In almost every depiction of Jesus, he is seen as some sort of Fabio model with flowing long hair. But James Caviezel probably should have kept his short haircut if he wanted to better portray Jesus. Not only was a short cut more in style at that time, but Paul writes against long hair (long hair was spurned by Judaism):
1Co 11:14 Does not even nature itself teach you that if a man has long hair, it is a dishonor to him?
I would say that the popular “recreated Jesus” is much more likely than the Hollywood pictures.