hi,
the 2 reasons that come to mind right away are 1st a folding prop will do just that fold on landing for planes that are hand launched and belly landed, then when used in glider type planes when you pull the power off to glide it folds back and will have much less drag in the air giving longer flights for those types of planes.
non folding props are stiffer and typicaly give you a little more power/thrust and are mostly used on planes with landing gear.
hope this helps some, im sure someone else can give better answer than I but its a start for you.
bassman