Is there an advantage of accepting one over the other. If so, why? Thanks.

There used to be no real advantage until over the last 20 years when people learned they can make fraudulent cashier's checks with thicker paper and a decent printer. You'll notice they are no longer considered "guarateed funds" when you go to make a deposit with one. A certified check may be a little more difficult to create from scratch because you are using a customer check that has had a red stamp on it and holes punched in the bottom. However, either way, the institution that accepts the check for deposit could call the bank it is drawn off for verification.

Both certified and cashier's checks are created for fraudulent reasons. Banks actually handle them much they way they always have and I see no advantage of one over the other, though cashier's checks are more common and may be easier to verify. It is still drawn on the bank and seen as guaranteed funds. That said, as was noted above a bank receiving one is within its rights to verify it.