Grab would be used to mean seizing something forcefully. It's synonymous with 'snatch'.
Grasp means to hold onto something tightly.
Both has somewhat less literal meanings:
Grab is also used to mean causing excitement or interest; 'The book was O.K., but it just didn't grab me' (from dictionary.com)
Grasp is frequently used to mean understand, or comprehend; 'I don't grasp what you just said.'