As verbs, all three of them are synonyms and very similar in meaning. All have to do with holding something tightly in one's hand.Originally Posted by AUTOMOON
In addition, "grasp" has a mtaphorical use. It can mean "understand" or "comprehend" as though one was grasping something with one's brain instead of one's hand.
"Clasp" can also mean "to hug or embrace" someone, or to fasten something with a clasp.
"Grip" can also mean "to hold someone's attention". It is also metaphorical, as if something is holding one's attention in a hand.