Answer:
A poem is poetry and a short story is prose.
The strictures and structures of the two forms are very different.
Poetry can be long or short - but traditionally the lines of which it is composed (like music) must follow certain patterns. Line length, syllables, accent, rhyme, and placement on the page are all crucial. Tempo, where the accent falls, how many beats per line. Imagery is often used.
Poetry is sometimes defined as concentrated prose.
None of those are considerations in prose.
A short story is a story that is told using a certain amount of text. Some are extremely short - less than a page - and others can be rather long.
A short story usually gives us a glimpse of a character, a slice of life. A single incident. Perhaps a man loses the overcoat he worked hard to afford and now accosts those wearing similar ones.
Perhaps a man remembers his lost love and mourns her passing.
Rhythm and imagery can appear but the focus is on the story told.
Poetry goes for a greater emotional payoff than prose and usually covers topics such as death, betrayal, love, loss, despair.
The how is more important that the what or who. Style lets the poet prove his mastery by making restrictions seem like natural pauses.
Sometimes, a line of poetry seems more line prose - as in:
When ten small words creep into one dull line
But the beauty of the phrasing is normally what sets it apart. Not just that it follows the rules of a ballad, stanza, haiku, or quatrain - but that the rules seem to embrace the work.
What is low, raise and support, what is dark, immune.
The One remains, the many change and pass
A rose-red city, half as old as time
In Xanadu did Kubla Khan a stately pleasure dome decree
where alph, the sacred river ran, through caverns measureless to man, down to a sunless sea.
I am not attributing these quotes in the hope the querent might look them up and read some great poetry.
Once you read poetry, you know the difference. Even with free flowing modern works - the word choice and placement is always deliberate and vital. Poetry sounds different, is more enjoyable when read aloud and yet strikes a very deep and personal chord in the reader when done well.
The minstrel wove his magic - worlds dissolved and fell, reappeared and rose in the shining mist
Some poets include dialogue and descriptions - but they are still done according to the rules of the art form.
Poetry can convey the same ideas and situations as prose - just in a very complex, subtle and concentrated way.
A short story is concise and word choice is careful, but not to the degree of poetry.
Prose can be translated with some loss of meaning, but still be enjoyable in translation.
Poetry, if translated, must be translated by a poet who then tries to preserve the soul of the original in his translation.
You really should read poetry - you will like it. In Tennyson’s Charge of the Light Brigade, you can hear the hoofbeats of the horses heading for doom. In his Charge of the Heavy Brigade, the sound of larger horses is clear.
There can be great depth and complexity to a poem and it often requires several reads to explore the various facets of this gem you experience.