See How and Why to Signify (July 2011) for some ideas about literary art in general.
It does little good to ask poets themselves. They are, after all, generally not critics, and their answers are likely to be fancifully impressionistic. Dylan Thomas, for instance, said “Poetry is what makes my toenails twinkle.”  He was trying to accommodate an interviewer, but it was in private conversation that Emily Dickinson said, “If I read a book [and] it makes my whole body so cold no fire can ever warm me, I know that is poetry. If I feel physically as if the top of my head were taken off, I know that is poetry. These are the only way I know it. Is there any other way?”  As we can be quite sure Thomas’s toenails never actually twinkled and Dickinson knows no more than we what it feels like to have one’s head blown off, what they were really providing was a sample of poetic language rather than a definition.
The question seems simple to those with a shallow knowledge of the subject. The most heavily used internet reference, a favorite of students, says simply that in poetry “language is used in a manner that is felt by its user and audience to differ from ordinary prose.” This would qualify a telephone book, I should think, or a physics research report sooner than a passage in Milton. Often definitions seem to consider all poetry as lyric, mentioning such qualities as short length or personal content – but what about epic? Or emotion, though some poetry is highly intellectual, witty, or centered in abstract form. Is there to be a quality standard? Is poor poetry necessarily not poetry at all?
Of course, etymologically the word simply means a “made thing,” an artifact if not an objet d’art, something shaped by human consciousness. In the Middle Ages the word poetry referred to all literature, that is, to aesthetic texts, whose consumers experience imaginative pleasure which is likely to be formal, emotional, and intellectual all at once. This classification is distinct enough to be useful, but it is not inherent in the work. Such pleasure can only arise in the reaction of the reader or listener susceptible to the text. Obviously some texts are far more likely than others to elicit an aesthetic response, but none are certain to do so for every consumer, and some critics may find aesthetic values unsuspected by an author or by earlier readers. 
What, then, marks off poetry from prose in modern usage? To note simply that it is laid out in lines of irregular length comes far closer to the mark than most answers and would have worked in a rough sort of way until quite recently. To explain how exactly the line breaks function is considerably more challenging, but verses serve very well as a sort of field guide identification key. The good sense of this criterion is not affected by the contested territory on the margins: on the one hand “prose-poetry” and, on the other, art-prose and rhyme-prose.  Still, as this descriptive attribute has so little theoretical value it remains unsatisfying.
I contend that, apart from the line breaks and the distinct traditions that have developed since the distinction emerged, prose differs from poetry only in degree. For instance poetry is musical. So is some prose. In 1945 John S. Barnes published a book of passages from Thomas Wolfe set as poetry, and recently David Amram has used passages from Kerouac’s On the Road as song texts. Poetry can still be said to be more likely to be more musical. In the same way poetry uses figures of speech and thought. A prose work such as Lyly’s Euphues or Philip Sidney’s The Countess of Pembroke’s Arcadia is heavily figured, yet in general it is true that poetry makes more frequent use of such devices than prose. A similar pattern obtains with all the frequently-cited characteristics of poetry.
For me this issue of poetry then turns out to be a matter of probability. Now that some scientists speak of reality as waves of probability this should, I suppose, seem an adequate, perhaps a superior, answer.
The cognitive argument of poetry, the modern form of the old notion of the poet-prophet’s privileged access to truth, is the claim that images are more precise than syllogism, and metaphors the most factual of data. To that deeply-held conviction I will add the anecdote of Robert Creeley who, as a onetime student recalled, entered the first session of his Modern Poetry class and began the class with a story. He explained that during World War II the Naval Air Corps had been having difficulties with fliers in the Pacific. It seems the pilots were frightened, not so much of Japanese adversaries, but of sharks once they hit the water. After the Navy began issuing canisters labeled shark repellent, they found their men had more confidence, greater success, and fewer casualties. The fact that the “repellent” had no effect of sharks did not mean it did not work. “Modern poetry,” said Creeley, “is like shark repellent.”
1. From “Notes on the Art of Poetry.”
2. Quoted in T.W. Higginson’s letter to his wife of 16 August 1870.
3. Prose poetry flowered in nineteenth century France though there are isolated earlier examples. In antiquity the borderline between prose and poetry was ill-defined and artful speech included highly poetic “Gorgian figures” and even rhyme-prose. For Aristotle, good prose is rhythmic (Rhetoric 3.8). Rhyme-prose was a popular form in the Latin Middle Ages.
4. Many readers of Gibbon, Freud, Marx, and James Frazer seek literary rather than scientific value. One might think also of secular appreciators of religious art or of Hayden White’s treatment of historians as rhetoricians.
5. From “Heaven's Commonplace: Hoc Opus, Hic Labor Est: Remembering Robert Creeley” by Donald Revell in Poetry.
Readers might enjoy having a look at the following exercise, written for a workshop.
Is this poetry? -- Why or why not?
1. Dad a dad a
Dad a da a
Dad a dad a
Da kata kai
Australian aboriginal song
2. If we shall suppose that American slavery is one of those offenses which, in the providence of God, must needs come, but which, having continued through His appointed time, He now wills to remove, and that He gives to both North and South this terrible war as the woe due to those by whom the offense came, shall we discern therein any departure from those divine attributes which the believers in a living God always ascribe to Him?
Lincoln’s second inaugural address
3. I'm Chiquita Banana, and I've come to say
Bananas have to ripen in a certain way.
And when they are flecked with brown
and have a golden hue,
Bananas taste the best, and are the best for you.
Original 1944 advertising jingle
4. “You and me down like four flats on a Cadillac.”
Prison inmate, 2002
5. I like Ike.
1950s political slogan
6. This style seems wild
Wait before you treat me like a stepchild
Let me tell you why they got me on file
'Cause I give you what you lack . . .
“Louder Than A Bomb” Public Enemy
7. Where wast thou when I laid the foundations of the earth? declare, if thou hast understanding. Who hath laid the measures thereof, if thou knowest? or who hath stretched the line upon it? Whereupon are the foundations thereof fastened? or who laid the corner stone thereof; When the morning stars sang together, and all the sons of God shouted for joy? Job 38, 4-7
8.The boy stood on the burning deck
Whence all but he had fled;
The flame that lit the battle's wreck
Shone round him o'er the dead.
“Casabianca” by Felicia Dorothea Hemans
9. “You’re the most beautiful in the world”
anonymous impassioned lover