I am not at all gastronomically inclined, but I figure that my understanding as a layman matters: "supă" is the clear stuff (maybe with some noodles, dumplings or small pieces of meat thrown in). It's also everything that is "instant soup" (e.g. packages of "dust" that you pour in boiling water).
"ciorbă" is the one with a bunch of stuff in it, such as vegetables and meat.
On wikipedia the difference emphasizes added acidity.