Why does Dade let the woman go at the end of the story?
This is a mystery. Here is my opinion.
During the whole story, Dade is going back and forth between his father and mother. His father is a practical man, who is responsible. He cares for his family and his business. Dade's mother is a dreamer, and she wants her son to dream as well. She wants him to look up into the skies and find something extraordinary. She wants him to do something great, like Thomas Edison (whom she mentions)
Because his father is giving him pressure, Dade makes a resolve to do something. What he will do is to capture the shoplifter to help the business. If he does this, he will be on the path of becoming like his father.
After he makes this resolves, he actually captures the shoplifter, to everyone's surprise. However, he has a change of opinion. Does he really want to be like his father? Does he want to dream like his mother? He is confused, and so he does not make a decision. He lets the woman go.