When we think "detective story," we think of something like Law and Order: SVU or The Mentalist or Cold Case, where the point is to put together all the clues to catch the perpetrator before he can kill again. Those stories are all about judgment. Society can't tolerate a killer in its midst, so we have the police (or the psychic, or the medical examiner, depending on the franchise) track down evildoers.
We don't know if it's because Poe wrote the first detective story or what, but "The Murders in the Rue Morgue" isn't really like that. By choosing a non-human killer, Poe neatly sidesteps the question of "justice" (because an Ourang-Outang isn't going to learn any better). And Dupin himself is not particularly interested in figuring out the case until a gentleman's obligation comes into it. In other words, he has his own code of honor, but he's not doing this for abstract reasons like protecting society or upholding the law. Neutralizing the question of justice gives Poe more of a license to turn his story into a puzzle. There's no strictly moral lesson to be learned here, unless it's that creativity is good and leaving your windows open and lights on at 3am is bad.
Dupin's decision to take on the case to repay a favor from Le Bon shows that he believes more strongly in honor than in justice.
By making the killer of "The Murders in the Rue Morgue" an ape, Poe shifts the task of the detective away from moral judgment and towards pure logic.