First, we need to understand between prose and verse:
Prose is the normal language that we use in normal speech, in literature it's the way most of novels are written using paragraphs.
Verse tends to be written with rhythms and consonances, instead of paragraphs it uses stanzas.
Now, the short story could be a small novel written in prose using this common method in modern literature that comes from short tales to a scientific book, while a poem is written in verse, often used in poetry and songs.