This is actually a question that poets and fiction writers deal with a lot, and it's fairly complicated, especially if you throw in the hybrid form of "prose poems."
If you just look at form, poetry is organized in stanzas rather than paragraphs, and emphasizes the line over the sentence. That's fairly obvious.
Then there are the simple elements of sound, which can appear in prose, as well, but are more common in poetry. This includes rhythm, rhyme, and meter, the latter two of which are not found in standard prose.
Poetry can indeed be more of an experience of "feeling" than some prose, but not always: consider the form of the epic poem, which relates a narrative, in contrast to the lyric poem, which is usually more (self)-expressive.
The best practical definition I've found in my writing career is that poetry, unlike prose, says that which cannot be summarized, and, in a truly excellent poem, cannot be expressed in any other way than the poet has chosen to phrase it.
For questions like these, it's really useful to consult dictionaries devoted to literary terms, as most standard dictionaries will not elaborate on this difference for you.
Hope this helps!