A cabinet surrounding your built-in barbeque grill adds useful counter space during prep, cooking, & serving, while under the counter space provides handy storage capability.
Expanding the design allows room for all sorts of other additional features such countertop side burners, refrigerators, shelving, storage drawers, etc. The possibilities are limited only by your imagination.
A grill island offers many of the same features, but comes pre-assembled and ready to just set in place.
Most of the BBQ brands we carry also offer grill islands.