Grab to seize suddenly or quickly; snatch; or to take illegal possession of; or to obtain and consume quickly: grasp To take hold of or seize firmly with or as if with the hand. or To clasp firmly with or as if with the hand. or To take hold of intellectually

"grasp" is more about holding something and emphasizes the hand. "grab" is more about taking it or moving it, though the first step might involve grasping with one's hand. The words also have distinct mental definitions. "grab" can mean cause interest in or gain one's attention. "grasp" can mean understand something.