Penalties often involve money. If one is late in filing a tax report, a financial penalty may be assessed. Whether or not this "hurts" a person depends on his financial situation and on how much he loves money.
Your question really raises a moot point, because as categories these wouldn't be negotiable. That is, you wouldn't be able to exchange a penalty for a punishment. On the other hand, the terms are used interchangeably in some circumstances.
You'd be more apt to be offered a choice of punishments than a choice of penalties. A penalty is usually intended to compensate the other party for some loss he's suffered because of your misdeeds. You illegally interfered with his play in some kind of ballgame, so he gets a free shot. You have to pay a thousand dollars a day because you're company is late in completeing some work on a building. This compensates the owner for his loss of business.
On the other hand, they often speak of the penalty phase in a murder trial. This I would consider a punishment issue.
If you plead guilty to a drunk and disorderly charge, the judge may say, "five hundred dollars or three weeks in jail." Different people in different circumstances might have their own reasons for choosing one or the other.
There was an interesting case where an American teen-ager was sentenced in Singapore to a severe caning for writing graffitti on public walls. This is obviously punishment in the purest sense.
Sometimes lawbreakers are punished by jail sentences and must also pay restitution to people who suffered losses because of their actions. I suppose you could say this involves both punishment and penalty.
One may speculate about personal obligations. You sometimes hear, "If you don't pay me the money you owe me I'm gonna take it outta your hide!" - meaning "I'm going to punish you - beat you up." Some people would pay a lot of money to avoid pain. Some people would endure a lot of pain to keep their money.