First, I think that the theme of this short story is that different people are different, no matter how much they like one another. In the story, the two men were like brothers and grew up in the same context doing the same things. But Jimmy went on to become a policeman and Bob a criminal.
The point of view in the story is that of a third party narrator, but one who does not see what is in the characters' thoughts -- they speak for themselves.
To me, the point of view in the story helps to reveal the theme because it does not give us any clues as to how the story will end. We are forced to try to figure out for ourselves what is going on. If the narrator were omniscient, we would know more about the characters and the suspense would be lost.
After Twenty Years derives much of its impact from the way in which it is told, and notably from the author's ability to fuse several points of view and to join tales from periods years apart into a single narrative with its own internal logic.
The style, while often richly descriptive and evocative, is terse, and the mannerisms of the two protagonists who tell their own tales blend imperceptibly with the narrator's anecdotal approach which would not have been possible but for the point of view, hence the importance.