At the various foreign ports, you need only a photo ID and your KTTW card (these in order to get off and re-board ship.) The Birth certificate/passport is needed at Port Canaveral to board initially, and it makes absolutely no difference which you have.
The idea is that you need one "official" government issued ID (thus, the certified birth certificate) and one photo ID (the driver's license). The passport has both features, so serves as both functions. Under 18 need only a certified birth certificate, although if a child has a passport, it is certainly accepted. Don't bother getting passports if you are a US citizen and don't already have one.
Do note that the birth certificate has to be "certified" This is the official certificate issued by a state government. It will usually say "certified" and/or have some sort of "official" seal (usually raised), holographic imprint, or other official sort of thing. What is not acceptable is a souvenier certificate issued by a hospital.