The reason using the Art Mapping feature of 3D Effect works for the map, but not for the hexagons of the soccer ball is: The map artwork is a cylindrical projection; the soccer ball artwork is not. It could be done by including the soccer ball pattern in the same Symbol as the map if you carefull drew the soccer ball pattern as a cylindrical projection.
This is exactly why 3D Effect yields unsatisfactory results in another recent thread about trying to map a patern of randomly distributed dots upon a sphere using 3D Effect. The resulting surface which 3D Effect generates in a revolved semicicle is not a patchwork of polygons; it is a single surface which necessarily must "pinch" rectangular artwork mapped to it at the poles.
When you say you want the soccer ball pattern to "revlove" around the globe, what do you mean? Are you building a singlle static image, or are you trying to build frames of an animation of a rotating globe?
If you're just building a single static image, your situation should be rather trivial to solve. You already have the soccer ball drawn, and you already have the globe drawn. You just need to overlay the two pieces of artwork and decide how you want them to interact, which is not clear in yohe ur description.
Is the map artwork raster or vector?
Is the soccer ball artwork raster or vector?
Do you want the final artwork to be one flat color? If so, what do you want to occur when the black of the soccer ball pattern intersects the black of the globe pattern? Do you want the intersecting areas to reverse to white? Do you want the soccer ball pattern to only show in the white areas of the globe pattern?
If you're building animation frames, and it is satisfactory for the globe pattern to rotate underneath the static soccer ball pattern, that would be similarly trivial.
If you're building animation frames and want the soccer ball pattern to rotate with the globe pattern as it spins, that is not so trivial, but can be done by redrawing the soccer ball pattern as a cylindrical projection and then including it in the mapped Symbol.