What is the difference between the verbs "grab" and "grasp"?
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.'