I am pleased to get to teach literature; it is where my training is, and much of my love. When I do, I run into the need to teach the three great forms: poetry, prose, and drama. When I do so, I inevitably field the question of what differentiates the three. It has taken me a while to find an answer that makes some kind of sense. It probably is not a great one, but it is one that works as an at-least introductory measure, something to get my students started on their literary study (which the students this semester have been doing well). Simply put, it is this: poetry organizes meaning by the line, prose by the sentence, and drama by the part in the performance.
Poetry works in lines. This will be obvious to any who have ever looked at or heard a "poet voice" recitation of one. Some poems do work in sentences, as well; Milton's "On the Late Massacre in Piedmont" comes to mind. Many work in stanzas, in groups of lines. But all will develop individual units of meaning in the individual line; it is at the level of the line that interpretation of a poem must begin. It is from the construction of the lines that poetic form is determined, whether the stringent demands of sonnets such as Milton's or the alliterative half-line Sievers types that typify Anglo-Saxon poetry or the rambling and erratic free verse that appears in much of my own verse and of my contemporaries'. (I sound remarkably arrogant, don't I?)
Prose works in sentences. This is something drilled into people nearly as soon as they begin formal schooling. We are expected to write in complete sentences in the main, told "a complete sentence is a complete thought." It is overly elementary to take past the fourth or fifth grade (although it persists among many who ought to be far past it), as there are complete thoughts that do not need "a complete sentence," and there are some that need several sentences to express in full. But orienting meaning primarily to the individual sentence remains the typifying feature of prose. There is just as much to work with in prose as in poetry, then, although the poem tends to be easier to find things to pick at than the prose. If nothing else, it tends to be shorter.
Drama works in the performed part. The chief distinction is the method of delivery; poetry and prose are determined and interactive between reader and text. Drama is interactive among reader, text, and audience; the addition of the third quantity (or the partial separation of audience from reader, for the reader remains an audience of the dramatic text) alters the construction of meaning and so alters the meaning itself. And it is in the interactions of the characters whose dialogue and actions are presented in the text that the drama organizes its meaning, whether that dialogue is in prose or verse (editions of Shakespeare, the playwright superlative, tend to show both) and whether the stage directions are explicit or implicit.
I admit that the definitions and explanations are clunky and provisional. They cannot help but be; they derive from what I have observed, and there are limits upon what I have seen and heard. Too, working from observation always admits of outliers and errors; there are things that trend away from center. The definitions and explanations are also meant to be unfinished; leaving them in such a state invites consideration and refinement by those I teach. As they work upon what I give them, they change it, and in expending the effort to change it, they make it their own. They learn it more than if I simply handed them refined things that I do not really have and would have to synthesize (and we know from food the perils therein). And that makes for better teaching.