The difference between poetry and prose is, to my mind, is one of communication. Not so much what is communicated, but how.
In prose, the main medium of communication is the content; the structure is there mostly to help the reader understand the content better. You use words with relatively concrete meanings which your audience is expected to know, and you fit them into a grammatical structure your audience is familiar with. This heightens the ease of communication and removes unnecessary ambiguity. The rules allow you to write what you mean and mean what you write; if you do a good job your reader will know exactly what you're trying to say. Prose is a representation of your conscious thoughts, altered to a form other people can more easily understand. I like prose; I use prose, I think everyone should use prose if at all possible. I recognize, however, that some things cannot.
In poetry, the structure has some purpose other then to clarify the content. At the very least poems use structure to make the content more appealing or attractive, but this only touches upon the true potential of poetry.
Prose is great, it allows you to put your thoughts in nice little boxes for easy transport. Unfortunately all thoughts do not fit in nice little boxes; language is ofttimes a poor tool at best to express the whole of human experience. The structure which prose follows is designed for clarity, not depth or breadth. By playing around with the rules of language one can enhance the scope of communication, though at the likely sacrifice of ease of understanding.
Some poetry is more structured then prose. This additional structure forces the author to be more creative; to find ways of saying things which do not disrupt the flow. Further, the structure provides a path for the reader to follow outside the flow of the story; a good poet can achieve interesting things by playing the flow of the content off the flow of the structure. Other poetry has less structure then prose. Without the clues which grammar usually provides, a poem becomes a much more ambiguous thing. A good poem is really many poems, depending on how it's read.
Most poems, however, use a different structure then prose which has more form in some ways, less form in others. This allows the structure itself to be a means of communication as well as the content. Phrases can play off each other in ways they cannot in prose. Elements of a sentence can break out of the classic subject/predicate dichotomy. Rogue line breaks and punctuation can disrupt the flow of the reader's thoughts and force them to look at the things in a new light. Words can be distanced from their traditional meaning, allowing both writer and reader to bring in new meanings.
Prose is usually best for communication from one conscious mind to another. Poetry, on the other hand, allows for deeper communication (from one soul to soul... or perhaps just deep structure to deep structure), perhaps). Thoughts, emotions, and feelings can all be transmitted through both prose and poetry, but prose depends solely on the standard meanings of words to do so. This is fine, as long as ones thoughts, feelings, and emotions are covered by the standard meaning of words. If not, you'll have to resort to poetry to communicate. Through the combination of words and structure you can perhaps convey what you want to others, though your chances of being misunderstood have increased.
Really, though, this is not such a bad thing. Being misunderstood can even be a good thing. Most prose has only one (or at most two or three) meanings. Writing a piece of prose is certainly a creative effort, but reading one is tends to be only a passive (if hopefully entertaining or informative) one. It should inspire thought, but more often only the thoughts the writer intended. Reading a piece of poetry, on the other hand, requires a lot creative effort. One must interpret the artist's words though ones own thoughts and experiences, which often generates new ideas that the author never intended. Hence it is a doubly creative process.