What is the difference between the ransom theory of atonement and penal substitution, and is either correct?

They are both wrong. The great irony of this topic is that our faith is based on the atonement, yet if there is one subject most people do not know how it works, it is the atonement. We say Jesus died on the cross for my sins, but what does that mean? What did the death actually do? After he died he rose again three days later; I see the result of that in the resurrection, but what did the death accomplish? The fact that Jesus obeyed the Father where Adam disobeyed does not itself save us; it only kept him pure, because God the Father wants a pure, sinless sacrifice, just as in the Old Testament the animal could not be mangled or missing anything. That only got him to the point where he could be the sacrifice. Some think Christ atones because he suffered so much, as in Mel Gibson's movie, but that is not it; some saints said one drop of his blood would have been enough, and to be frank, no. What was required was not the suffering but the death. Unless Christ died there is no atonement, no matter how much blood was sprinkled on the way; the suffering is only an artifact of the real reason. The ransom theory says the devil is owed something and God must pay the devil with the life of Christ to free the slaves under him. No. The devil has nothing to do with it; he is under judgment and is lucky God did not destroy him outright. The Greek word translated 'ransom' misled people; it does not mean God had to give the devil something. Penal substitution is the idea that Christ was our substitute who paid the penalty for our sins so we go free. But Christ did not pay for our sins on the cross, and the Church never taught that. John Calvin taught it because it fit his theology: he denied free will, so God arbitrarily chose the elect, and their sins had to be paid, so Christ had to pay that price. Yet dying on the cross, as terrible as it was, is not hell, and he was out of it in three days; there is a theological imbalance. Calvin tried to fix it by saying the punishment began in Gethsemane, that this was the 'descended into hell,' but that does not work. If Christ actually paid for your sins, then by double jeopardy nobody, not even Judas, could go to hell, and the pope would be right. That is not the atonement God gave us. To understand it you must understand who God is: not a mechanical cause-and-effect being as Thomas tends to describe, but a personal being with love, compassion, pity, and honor, who said he was sorry he made man before the flood, using the strongest Hebrew word for sorry. When you sin you personally offend him and besmirch his honor, and he has every right to destroy you. What turns this mighty being back toward you to offer forgiveness? A sacrifice like his Son offered on the cross, a pure, sinless God-man who offers his life to restore the Father's honor and appease his wrath. That is all the Father will accept. And then he does not send you to hell to pay for your sins; he forgives you and gives you the same chance Adam had, though now under concupiscence and in a fallen world. So it is neither penal substitution nor the ransom theory; the atonement is personal, not a mathematical or legal calculation.