What are some concrete differences between a "short story" and a "narrative essay"?
A general criteria is that a narrative essay describes a story from a personal point of view, with a special emphasis on a "point" or "lesson" conveyed by that story. However, this is hardly a reliable distinguishing criterion. A short story may also have these features, making it difficult to distinguish it from a narrative essay. So what are some other criteria that can help make the distinction clearer?
An "essay" is necessarily non-fiction - a discussion of facts.
A "short story" is a short piece of fiction; unlikely to be more than 8000 words in length.
A "narrative essay" uses your personal experience and expresses it in a manner that is close to telling a story.
This seems a good clear description: Essay Info :: Narrative essay writing