In "A Retrieved Reformation," why does Ben Price decide to let Jimmy go?

The reader feels gratified that Ben Price decides not to arrest Jimmy Valentine. This must be because the reader feels Jimmy should be rewarded for saving a little girl from certain death. Ben Price is not quite the same kind of law-enforcement agent as Jimmy Wells in O. Henry's "After Twenty Years." Jimmy Wells is a sworn officer of the law and is duty-bound to arrest 'Silky' Bob, or at least have him arrested, after he recognizes his old friend as the man who is wanted by the Chicago police. Ben Price, on the other hand, is a sort of private detective. These were the days before the federal government began insuring banks and thereby got involved in that aspect of law enforcement. Ben Price can arrest Jimmy Valentine or let him go, as he chooses. Ben acts on the basis of the same feelings he shares with the reader.

We see in the Sherlock Holmes stories, such as "The Adventure of the Blue Carbuncle," that Holmes as a private citizen can and does let criminals go free. Ben Price feels the same way about Jimmy Valentine's act of sacrifice as does the reader. 

“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.”

And then Ben Price acted rather strangely.

“Guess you're mistaken, Mr. Spencer,” he said. “Don't believe I recognize you. Your buggy's waiting for you, ain't it?”

And Ben Price turned and strolled down the street.

It should be noted that Ben apparently knows that Jimmy has gone straight, that he is getting married to a respectable girl, that he is abandoning his burglar tools, and that he must be giving up his life of crime. So Ben Price has at least two reasons for letting Jimmy go free. One is that Jimmy deserves a great reward for saving the life of a little girl at the risk of losing his freedom and all his claims to happiness. The other reason is that Jimmy is no longer a threat to society. He won't be cracking any more safes, so Ben no longer has a strong motive for taking him out of circulation.

How does Ben Price know Jimmy in "A Retrieved Reformation"?

Ben Price is a highly experienced detective. He's known about Jimmy Valentine for years. Jimmy's an infamous safe-cracker, and he has a very specific modus operandi, or way of working, that's become his trade-mark. Years before, Ben had arrested Jimmy in a room above Mike's restaurant, the room where Jimmy returns after his release from prison. Ben's certain that since his release, Jimmy's been up to his old tricks again. At the scene of his latest crime, all the old signs of Jimmy's handiwork are there: it was a clean, easy job, and only Jimmy Valentine could have used the tools necessary to crack open this safe.

Ben Price clearly has a good reputation as a detective. We know this because the narrator tells us that people with safes full of money were glad to hear that Ben Price was at work trying to catch Jimmy. Although Jimmy refers to Ben as "my old friend" in the letter he sends him, the relationship between the two men appears to be strictly professional.

How does Ben Price save Jimmy in "A Retrieved Reformation"?

Ben Price thinks that Jimmy is still a criminal and wants to arrest him until he saves the little girl, and Ben does not tell anyone who he is and lets him go.

Jimmy does plan to rob the bank, until he meets the love of his life.  Upon seeing her, it is love at first sight for him and she seems to reciprocate because he is so handsome.

A young lady crossed the street, passed him at the corner and entered a door over which was the sign, "The Elmore Bank." Jimmy Valentine looked into her eyes, forgot what he was, and became another man. 

Jimmy changes his name, becomes respectable, and begins to court the lady.  She just happens to be the daughter of the banker, so it makes sense to suspect him of being up to no good.  Alas, this is not the case.  Jimmy risks his new life and his freedom to use his safecracking skills to rescue his only love’s little sister when she accidentally locks herself in the safe.

If Jimmy had not acted, the girl would have died.  Jimmy was sure that acting meant getting outed and arrested.  He would lose his love, and be locked up.  Yet Jimmy’s transformation was genuine, and he would rather risk anything to spare his love pain.

Ben Price realized what Jimmy had done, and the risks Jimmy took.  He also knew why he took them.  He decided to pretend he did not know Jimmy and go on his way, leaving him to a new life as a reformed safecracker.