I don't believe there really is a distinct denotative difference between the two. For example, one may insist that a short story requires a plot, while a poem needs have none. Yet it may be possible to create a short story that is missing various basic elements of the plot structure: no clear beginning (something told "in medias res," or in the middle of), no clear climax, no clear epilogue, etc. You might try to define a short story as needing one or more characters, yet again I think one can create a story that defies that definition.
I think the clearest distinction between the two lies in the arrangement on paper, as well as general tendencies characteristic of each form.
A poem is written in lines which are not indented as a regular prose passage would be. These lines may or may not rhyme and they may or may not have a syllabic pattern. They are commonly brief (a page or less) but are not required to be (i.e. Dante's Divine Comedy, which is 100 chapters long). They typically take some poignant thought, emotion, or urgent communication of the author and give it voice through very particular word choice, attention to sound, arrangement on the page (i.e. enjambment), etc.
Short stories can employ poetic devices, and poems can employ narratives, tell stories, etc. The line between them is grey. Again, ultimately, the difference lies really in the arrangement on the page and not in the subject matter.