I can only give my opinions but here is what I perceive the differences to be:
"Traditional" applies to the tried-and-true old patterns that have been done for decades, if not centuries.
"Contemporary" means updated for today. It can include the traditional patterns done in updated fabrics and settings.
"Modern" is a category of patterns and fabrics that departs from all the form and rules of traditional quilting. Anything goes.
"Art Quilting" is more about creating a work of art than a blanket. Art quilting is usually quite labor intensive and is quilted very densely, making a great item to hang in a gallery, but not so much to cover yourself with for warmth!
"Fiber Arts" is a very broad category that all these styles of quilting fall into, but so do so many other things-- any and everything created with fiber of any kind.
These are only my layman's understanding of these terms.
I can't really speak to the other things you asked, as I have never participated in any kind of quilt show.
Last edited by patricej; 10-08-2013 at 04:01 AM.