In order for Ben Price to arrest Jimmy Valentine, he would have to believe Jimmy was not reformed.
Jimmy is a bank robber who was released from prison but seemed to have gone back to his old ways. Ben Price is a cop that is interested in justice, and wants to make sure Jimmy gets what’s coming to him.
The “retrieved reformation” in the title refers to Jimmy’s becoming a new man when he falls in love. He has to use his safe-cracking skills to free his new love’s little sister when she gets herself locked in a safe. Ben Price is so inspired and impressed that he chooses not to arrest Jimmy.
"Hello, Ben!" said Jimmy, still with his strange smile. "Got around at last, have you? Well, let's go. I don't know that it makes much difference, now." …
Guess you're mistaken, Mr. Spencer," he said. "Don't believe I recognize you. Your buggy's waiting for you, ain't it?"
There are different ways this ending could have changed. Ben Price might have decided that it was just a matter of time before Jimmy robbed a bank, and he wasn’t reformed. In that case, he could arrest Jimmy anyway. He might also decide that it doesn’t matter if Jimmy is reformed or not, because his duty is his duty. He is a cop, and Jimmy is a robber. It does not matter what Jimmy might have done, because all that matters is that he has committed crimes and needs to pay for them.
O’Henry is known for irony, and if Ben Price did arrest Jimmy, the story would have been quite different in meaning. It is part of the irony that Jimmy has to use his criminal skills in order to rescue the girl, but it is also part of the irony that Ben Price decides not to arrest Jimmy after all. Some of the irony would therefore be maintained, but the ending would not be quite as unexpected. Ben Price would be seen as the bad guy.