The prose poem is an understandably controversial form. It grew out of a reaction, perhaps, to the extremes of free verse as it was being written (and promoted) during the mid-20th century.
I am not making this up.
I've been asked, "What distinguishes the prose poem from free verse?"
This question will necessarily lead to a more extensive rewrite than I had first envisioned when I set out to fill this node-shell. However, to take a first stab at addressing the question, at least in my experience, it seems that free verse makes some effort to retain some element of the "look" of a poem. Usually this is done through line breaks or other formatting of the text on the page, conventions that, to the casual reader serve as a marker, saying "This is a poem."
By contrast, the prose poet typically tries to retain the "feel" of a poem, the heightened, semantic density of both explicit and implicit meanings and allusions, while intentionally avoiding such trappings as rhyme, meter, or other kinds of phonological and visual structure that are often hallmarks of even the most offensive sorts of doggerel verse.
While sharing many of the intellectual aims of free verse, to challenge preconceptions about what poetry is or can be, most prose poems look, at first glance, like an ordinary paragraph or two of prose. There are no forced line breaks, except for those one would find in a similar passage of prose, and the appearance of the poem generally follows the formatting rules of prose, as well as its lack of attention to rhythm, rhyme, assonance, alliteration or other "sound-based" (phonetic) tools of the traditional poet. The punctuation is functional and tends to stay well within the bounds of ordinary prose.
Usually a prose poem calls attention to its poetic content (or pretensions, or conceits) chiefly through diction, verbal imagery, or a heavier reliance on metaphor and other forms of figurative speech that are used more sparingly, as a general rule, in fictional narrative and in non-fiction, expository prose. Even in most literary fiction, novels or short stories, imagistic and non-literal patterns are rarely used, and are almost never as densely layered as they tend to be in a prose poem. The exceptions I can call to mind here are books like Ulysses or Finnegans Wake, and some might want to make the case that both of those may qualify in some ways, as Modernist Epic Poems.
Brevity is also a common marker that may tell you that you are looking at text that wants you to think of it as poetry and prose at one and the same time.
Academic discussions of these two forms seem, in general, to try to categorize the methods common to each one, as well as looking at the context in which examples of each form may have emerged.
Notes for Node Expansion:
If it were earlier in the day, both literally and metaphorically, I would paraphrase The New Princeton Encyclopedia of Poetry and Poetics on the subject. The article there under this heading is quite instructive. I will try to locate the one collection of prose poems that I was asked to buy as a creative writing student at Oberlin College in the late 1970s, and offer some names of poets who used the form to some interesting, even evocative ends. At least in my opinion they did.