You don't need passports, but they are designed to withstand the rigors of travel whereas (whereas - what a word and I'm not an attorney) birth certificates are "just" paper and much easier damaged.
That being said on all of our prior cruises we've relied upon the BC, but in the future we will be taking our passports.
You need your BC/passport to get on the ship in Port Canaveral. You just need your photo id to get off and on the ship in the ports. The one exception is on your stop in St. Thomas - you need the BC & photo id or your passport before anyone can get off of the ship (go through immigration).