Jimmy sends the note to 'Silky Bob' for the same reason that he cannot bring himself to arrest Bob when he recognizes him in the doorway of what used to be 'Big Joe' Brady's restaurant--they once were very close friends.
Some things do not wear away with time, and the friendship between Jimmy and Bob yet has a place in the hearts of both Jimmy Wells and Bob. For instance, when Bob speaks of his friend Jimmy to the policeman--who, ironically is actually Jimmy--he mentions what a "good fellow" he is as well as how faithful he is,"If Jimmy is alive on earth he'll be here by that time."
"Faithful friend" that Jimmy Wells is, he, nevertheless, must put duty first. For this reason, he goes to the police station. However, he does not have the heart to so embarrass and shame his old friend by arresting himself. Therefore, he asks a plain clothes policeman to make the arrest for him. Furthermore, he wants Bob to understand that he has not forgetten his promise to meet Bob twenty years later because he was at the doorway of their favorite restaurant on time. Because of these reasons, Jimmy writes his old friend.