Bob walks up the street arm in arm with the plainclothes detective, thinking the tall man is his old friend Jimmy Wells. When they reach the lighted drugstore Bob balks.
“You're not Jimmy Wells,” he snapped. “Twenty years is a long time, but not long enough to change a man's nose from a Roman to a pug.”
Jimmy Wells had a big Roman nose. This plainclothes detective has a pug nose. Evidently it had been the detective's intention to lead Bob arm in arm all the way to the station house. But at this point he is forced to admit that he is not Bob's old friend and to tell Bob that he is under arrest.
O. Henry wanted the arrest to be made as peaceably as possible because it looks bad enough for Jimmy to be turning his old friend over to the law, and it would look worse for Jimmy if there were a violent arrest involving a gun, handcuffs, and physical force. But Bob has to find out he is under arrest before they get to the station house. That would take up unnecessary time, and it would depart from the compact scene in which O. Henry has adroitly set the whole story. The lighted drugstore is like the extreme outer-edge of the setting.
The pug nose instead of the Roman nose is an unmistakable way for Bob to realize instantaneously that he has been tricked. He still can't try to make a break for it because he is locked arm in arm with the plainclothes detective, who is a bigger man. This is a good spot to end the story. They are standing right in front of a brilliantly lighted drugstore window which O. Henry has created and placed there for his surprise ending. The lighted window also provides a convenient place for Bob to read the note from Jimmy. He would have a very hard time finding another place to read it in this neighborhood. O. Henry describes the setting in the opening of the story:
Now and then you might see the lights of a cigar store or of an all-night lunch counter; but the majority of the doors belonged to business places that had long since been closed.