I recently did a "Boot Camp" with the goal of writing a flash fiction or poetry piece, per day, for two weeks. I mostly concentrated on flash because that was what I "needed" for submission purposes, but I did finish four poems. (I partly stopped with poetry because they were becoming increasingly disturbing ... not sure why that happened, but I needed to stop unnerving myself with my own writing.)
What I became aware of is that writing poetry, particularly - for me, at least - within fixed form and line lengths, helps strengthen a writer's sense of word choice. In a short story or novel, it's easy for a cliche phrase to slip by in the flow to the next and the next. In a poem, the content is finite and each phrase needs attention, and often reworking to arbitrary lengths or rhythm. This draws a writer's eye with laser focus to the exact words, the way of shaping image: the journey as well as the destination.
Some writers will also use a form as a starting point and depart from it when it doesn't serve them; this is another great way to heighten awareness of exactly how you're making your point. Is the original phrase(s), within the context of the form, most effective, or does this change that departs from the form enhance the poem?
Like flash fiction, a poem is also a way to crystallize an idea in a compact number of words. Finding that essence makes the writer aware of what's actually needed to convey the story. (And there is a story, even if it's a progression of moods or an internal conversation rather than a specific plot.)
As a writer, I tend to be fairly deliberate: if a word choice isn't right, or I'm missing a fact, I need to resolve that before continuing. I spend a lot of time in my initial write of story openings to make sure that all the pieces are entering play. People who toss in parentheticals to (fill this in later) boggle me. But even if one is more a "throw down words and don't look back" writer, poetry can be helpful when you get to the editing stages. Clunky or dull phrases leap out where it might be possible to skim past them in a manuscript.
I happen to write (usually) overtly fantastical poetry: seers, ghosts, aliens. But even if tackling more mundane subject matter, poetry sharpens focus and attunes one to specific word choice.
Word Count: 8,560
Poems written: 2
Pages edited: 5
Word Count: 4,857 (... it's been a week)