The main difference is the source used to generate the view: clipping views use the 3D geometry and the source view, while quick clipping views just use the source view. Here's a similar discussion regarding detail views: Detail and Quick Detail views in Drafting
My error, I should have stated "reference view" rather than "source view" to use the correct terminology. I'm just now realizing that this isn't applicable in a clipping view since there is no reference view for a clipping view. The basic idea of a plain clipping view referencing 3D and the quick clipping view referencing 2D only still appears to apply though: Creating a Clipping View / Clipping View Profile .
Practically speaking, I'm not sure what the difference is. I don't think I've ever used a quick version of any view, but I am also far from a power/experienced CATIA user.