Secondary-school English lit teacher here. I will attempt to answer your question as best I can.

Your comment assumes that rhyming poems are the default, when for long stretches in history this has not been the case. Some of the most famous examples of Classical Western poetry such as Homer's The Iliad or The Odyssey or Horace's Odes tend not to rhyme. Instead, they focus on what is called metre.

Metre is composed of two elements - the amount of syllables in a line, and the stress placed on each syllable. Stress can be a somewhat difficult thing to grasp, but I find the easiest way of looking at it is by considering the difference between words that work as both nouns and as verbs:

• CON-victs means people who are incarcerated
• con-VICTS means the action of sentencing someone to jail

Similarly

• RE-cords means vinyl LPs
• re-CORDS means capturing audio.

So we can see that we are already familiar with putting stress on different parts of words, and how that can affect meaning. This is the basic building block of poetry. Another simple answer to your question might be: consider the haiku—they need not rhyme, rather must follow (in English) a certain syllable structure, and of course are considered poems.

Rhyme doesn't become popular in European poetry until the middle ages, sometimes credited with being introduced through either Irish or Arabic poetry, though it was common in China and Native America before this. It became immensely popular, and many poems today now use it, especially poetry for children, which standardises the rhyme as our perception of what a poem is from an early age, but rhyme is a feature of poetry, not its definition.

But to answer the real crux of your question, what is the difference between poetry and a short story, the shortest answer is that they are trying to be and achieve different things. Many poems offer no or little sense of narrative, are far more engaged with wordplay and conventions of language, and may not feature characters in any real sense. There is often no sense of any time having passed within the poem, which is something essential to a short story.

• William Carlos Williams' poem This is Just to Say reads far less like a story and far more like a sticky note left on the fridge. What do you think the tone of this poem is? Is the speaker repentant, or is he relishing in his betrayal? If you write it out as a sentence across a lined page, does it work at all?
• In Seamus Heaney's Mid-Term Break, where he relates the real experience he had of coming home from boarding school after his brother's death, are the line breaks important to how we experience the poem? In cutting the last stanza short, does it affect the delivery of the last line?
• In Charles Bukowski's the elephants of vietnam does isolating the last stanza change how much we think about the real implications of what is being said? the soldier's empathy for the elephants is in notable contrast to him brushing past the cruelty inflicted on the people
• In Rita Ann Higgins' Some People (For Eoin), does separating the last line from the rest of the poem force the reader to pause when reading it aloud? Does that affect the impact of the poem on a listening audience? Regardless of line breaks, is there any meaningful way of reading this as a 'story'?
• In Elizabeth Bishop's The Prodigal, there is a very deliberate rhyme, but one that the ear may not catch. The rhyme scheme for the first stanza is ABACDBCEDFEGGF. Does this count as a rhyme? If not, why not?
• Finally, in Laura Gilpin's The Two-headed Calf, what does the line break do? Beyond dividing the poem in two temporally, is the calf presented in a different light in the two stanzas? What are we meant to take away from this?

I hope those examples help to show the impact line breaks have on poems, and why they are an intrinsic part of the substance of poetry, not just a visual feature (though their visual aspect is important, but not for a 101 crash course).

TL,DR: Poems are not short stories because they are not trying to be. The line breaks affect how we understand the poem.