Pretty much the same. Grab means to take forcefully ( before someone else takes it). Grasp means to hold on to something tightly. I have never heard anyone say "grasp" an opportunity.

There is a similar Latin phrase too 'carpe diem!' seize the day!

Grab suggests Sudden motion; Grasp is to hold and not let go. Grasp is the correct phrase, because 'holding on to an opportunity' that is what is meant by the phrase. Seize an opportunity is a more common variation I think

Someone making a formal introduction may "take the opportunity" and drone on endlessly. The person being introduced may "seize the opportunity" to make an early entry. Often, I grasp at straws when trying to learn another language. (Futile attempts)

It would be more common to say "to grasp at an opportunity" instead of "to grasp an opportunity". This would imply maybe that the person is taking a somewhat desperate chance at something. To grab an opportunity implies that the person is being active.