Working from Beryl Dean here, one of the more important vestment designers of the last century: At the end of the middle ages the cope unfolded into a semicircle and the orphreys (that is, the bands along the front edges) became increasingly wide and stiff. The result of forcing this into the shape of a cloak was a conical garment which the bishop stood inside almost like a teepee. At the same time the hood unfolded into a decorative flap. People who want to assert Tradition prefer that form, but the tendency has been for it to revert to being properly cloak-shaped and for the hood to be formed as such.
The chasuble is different: it is always formed with a hole in the center for the head. Again, after the middle ages the sides of the chasuble tended to disappear to free the arms for elevations, so that you ended up with the "fiddleback" form with a rectangle on the back and a cutaway shape on the front. The more modern form evolved separately from the Gothic form (which was indeed more or less conical but which was joined up the front, unlike the cope) and which tended to flatten out until it became a flat circle. There are names for the different chasuble forms, but not for the copes.