Russell Edson said, “A good prose poem is a statement that seeks sanity whilst its author teeters on the edge of the abyss.”
As for me, I haven't yet decided. Is there a difference in compression? in the predominance of figurative language? in the music of the language? Yes. No.
No end words:
Poetry yields as many end words as there are lines in the poem. End words are particularly important in poetry. They are usually very important: nouns or verbs. They can be used skillfully to develop enjambment and to increase meaning through a line break and ambiguity. Prose poems have but one end word. This may allow a poet to give the emphasis to the word that ends the prose poem. In microfiction or flash fiction, not much attention is paid to the end word of a paragraph. Maybe not enough attention.
Stanzas or paragraphs
Stanzas are poetry's paragraphs. Turns occur between stanzas. Leaps can be made in time and space. In prose, there are transitions, associations, and leaps but the paragraph might be barely noticed. It is easy to conclude that form is less important in prose. One still can use music, rhyme, rhythm, and sound patterns. They are "buried' inside the paragraph.
Form
The form of a poem might be considered a container. Poetic forms have history, patterns, associations: sonnets, villanelles, sestinas, quatrains, etc. Prose forms are more loose and flexible. A poem or a prose might be a letter, a list, a story. Content is perhaps more important in prose. The intention might be to entertain, investigate, or illuminate.
Audience
Perhaps the answer resides in which audience one wants. Naming the genre or using a genre form sets up an expectation. In prose, genres are sci-fi, fantasy, murder mystery, romance, western, or nonfiction. Poetry is a genre. Prose poetry might be blend of genres, a hybrid form. Of course writers of both poetry and prose explore and widen the boundaries. There is plenty of genre-bending.
Definitions:
According the Poetry Foundation, "A prose composition that, while not broken into verse lines, demonstrates other traits such as symbols, metaphors, and other figures of speech common to poetry. See the prose poem,"Information" by David Ignatow: https://www.poetryfoundation.org/poems/43416/information-56d222285400c
Perrazo and Dal wrote an academic paper examining the differences and similarities between these two forms. They identify criteria: brevity, fragments, ambiguity, and pivots. Either the prose poem or the flash fiction (or microfiction) could be narrative. The difference might be that the microfiction suggests a larger story or is a pivotal moment in a larger story.
Writers:
Amy Hempel has written brief fiction. "Sing To It," in the book of that title, is less than half a page. Of her writing, Amy Hempel said, “I leave a lot out when I tell the truth. The same when I write a story." In a review of Hempel's work, Ruth Franklin said, "For Hempel, a story often takes place not at the moment of crisis but in its aftermath." Hempel recalls a writing class with Gordon Lish. For his students’ first assignment, Gordon Lish instructed them to write about their worst secret: the thing they had done that, as he put it, “dismantles your own sense of yourself.”
Lydia Davis is a master of short fictions. "Letter to a Funeral Parlor" is simply that, a brief and amusing (and poignant) demand for an explanation of the use of the word "cremains." Davis cites the influence of Russell Edson. She is interested in stories, and she is interested in etymology. In her work, they often come together in ways that are comic and profound.
Diane Williams writes very brief narratives. They are elliptical, associational, and sometimes even musical. See https://lithub.com/bang-bang-on-the-stair/ and https://www.vice.com/en_us/article/mvx79p/five-very-short-stories-0000667-v22n6, plus http://www.thewhitereview.org/feature/diane-williams-two-stories-and-an-interview/
Diane Williams writes very brief narratives. They are elliptical, associational, and sometimes even musical. See https://lithub.com/bang-bang-on-the-stair/ and https://www.vice.com/en_us/article/mvx79p/five-very-short-stories-0000667-v22n6, plus http://www.thewhitereview.org/feature/diane-williams-two-stories-and-an-interview/
Peter Orner dislikes the term flash fiction. This form is not new, he says. "How many lines are in the story of Adam and Eve?" he asks. Well, I looked it up. Chapter 3 of Genesis is less than 700 words. Electric Literature has samples of flash fiction. https://electricliterature.com/7-flash-fiction-stories-that-are-worth-a-tiny-amount-of-your-time/
Nick Ripatrozone wrote an essay examining this question. He says, "...genres are not spouses. Monogamy has no place here: the writer should–and must–be flexible to genre. Genre, mode, form: these considerations are contextual and situational."