Two illustrations to help you:
1) Biblical atonement is like this: A Judge declares that you owe a debt of 1 billion dollars next week, or you will go to prison. You have no way to pay the debt. But the Judge offers to pay the debt himself on your behalf.
But penal substitution is different. According to penal substitution, the Judge sentences you to the electric chair. The Judge then volunteers to go to the electric chair instead of you, in your place, so you never have to. Penal substitution advocates will insist that Jesus paying a debt on your behalf and going to the electric chair in your place are equivalent in terms of the priorities of justice being satisfied. But that is not the case in the Bible. The Bible does not equate payment and punishment (restitution and retribution are distinct priorities of justice in the Bible).
2) Biblical atonement is like this: You crash into your neighbor’s car and do $1,000 worth of damage. Your neighbor is angry. Your insurance company, “Jesus Insurance”, pays your neighbor $5,000 on your behalf, and your neighbor is no longer angry (he is propitiated), he fixes his car, and he ends up with a surplus.
But penal substitution is different. According to penal substitution, you crash into your neighbor’s car and do $1,000 worth of damage, your neighbor is angry, but then your neighbor demands that he either do $1,000 of damage to your car, or the insurance company provide a vehicle of equal worth to his own to which he can do $1,000 of damage. His anger can only be propitiated if it is exhausted on your car or a substitute car of equal worth to his own. The neighbor doesn’t just want a debt of restitution, he specifically demands a so-called “debt of punishment” in which he can fully exhaust his anger (even though no such debt of punishment would actually fix his car).
According to penal substitution, Jesus pays our so-called “debt of punishment”. But in the Bible, there is no such thing as a “debt of punishment”, and what Jesus pays is our debt of obedience in order to make restitution to repair what was broken by our sin.