The short answer to this question is that rhyme is like a companion on a journey.  Inspiration is often like having a mountain come into one's contemplative view.  Oh, what a wonderful notion -- I should try to write a poem on that notion!  That's often how a poem is born or conceived.  If the notion is the mountain and you want to get to the top of the mountain, to express the essence of the notion, and you begin to write (one or two or three lines, experimentally), you can then begin to think about what you want to write next and if you can rhyme with the next lines in some pattern, one either traditional or even novel, and advance toward and with the essence of the notion (the top of the mountain vision), then in doing so you have a companion -- you have begun an inner dialogue with the Word.  It takes discipline to write with rhyme because you can not cheat yourself of the essence of your original notion and be led astray by rhyme.  Rhyme and a rhythmic flow of words add a certain pleasure to the poetic line.  But rhyme for rhyme's sake is meaningless.  In writing with rhyme one has to take, perhaps, more time with one's composition and weigh what can be done with a line to make it rhyme and keep it true to one's original inspiration.  Some of my poems, for instance, I have revised over the course of more than 20 years, for example, "I Read a Poem" and "Our Marriage Bed."  Both had lines that I never felt comfortable with, and I kept trying to find a better, more focused, and/or more lyrical way of expressing myself whenever I thought about these poems.  Seeking to improve a poem one already loves carries its own delight -- in the possibility of making them more wondrous and meaningful and alive.

Like a companion, rhyme can often surprise you, the poet, and introduce ways of getting to the mountain top that you would not have thought about if you were avoiding or careless of rhyme.  Alliteration can work this way as well and as does, certainly, limiting each line to a set number of syllables or rhythmic pattern.  These sorts of limitations, if embraced, may serve to enhance and provide dynamic structure to one's expression.  Somewhat like the steps in a formal dance pattern -- when carried off well such a dance may be far more moving than a spontaneous gaiety.  But one can never be a thrall to poetic devices because they will lead you to a meaningless end.   No matter how stirring the inspiration, one most always be willing to crumple up one's work and toss it aside if the original notion is being sacrificed -- even if it means doing so over the course of twenty or more years.