Why couldn't Jimmy arrest Bob himself in "After Twenty Years"?
At the end of the story, Jimmy sends Silky Bob a note that says,
Somehow I couldn't do it myself, so I went around and got a plain clothes man to do the job.
We can surmise, from everything that Bob has said about Jimmy, that Jimmy still feels a strong bond of friendship for Bob, even after twenty years have passed. Bob describes Jimmy as a fine, upstanding person, who unlike Bob, never wanted to leave New York. Bob calls Jimmy the "truest, stanchest old chap in the world." He refers to Jimmy as a "plodder," the kind of person who works slowly and faithfully at one job, rather than flit around to different criminal activities as Bob has done in a quest to get rich quickly.
Ironically, Bob's talk shows he feels he has been more successful than Jimmy. It also shows he has complete faith in this old friend. However, it is the very characteristics in Jimmy that Bob admires—loyalty, faithfulness, and integrity—that would lead him to put his duty to his job first and have his old friend arrested.
Jimmy's note suggests he is a person of heart and feeling. As much as he knows he has to do the right thing and get criminals off the street, he still remembers his old friend with fondness. Having to arrest Bob is upsetting to him.
The story is, on one level, a study in character. An appreciation of the kind of person Jimmy is pulls Bob back to meet with his old friend, but this appreciation is Bob's undoing.