Cayman IS great, but it can be more expensive and there are a few extra steps involved in getting in and out of the country... If money is no object, then rock on down to Cayman.
There is great variety and number of dive sites appropriate for your experience level in the Florida Keys that will keep you interested for a long time. The good thing is that you can easily grow your experience as you progress to deeper and different sites (wrecks, reefs, marine life encounters, etc.)
To each their own, but I have to disagree with k ellis. The Keys are a great family vacation spot. The place is mostly local flavor, which to me is the best part--I don't care for tourist spots with an Applebee's on every corner. Topside there are plenty of great activities like Theater of the Sea, History of Diving Museum, Islamorada fishing, Mangrove/Flats Kayaking, John Pennekamp State Park, etc, etc. not to mention all the standard beach activities like parasailing, jet ski rentals, great seafood, public beaches and more. Keep in mind, this is all middle and upper Keys--the lower Keys and Key West list would take pages...
But as far as DIVING goes:
Try Conch Republic Divers in Tavernier, FL--they are by far my favorite dive operator. Molasses Reef alone is about 20 acres so you can dive it for a week and just about see it all. French Reef is pretty large as well and progresses from 30 feet to about 70 feet on the southern edge. Everybody needs to see the Christ Statue at least once.
There are wrecks like the Benwood, that go from about 25 feet below the surface down to about 90 feet so you can dive as deep as you are comfortable with.
There are a ton of deeper wrecks in the Keys for when you are ready for more advanced dives.
Captain Slate's Atlantis Dive Center does a Creature Feature dive (I think on Fridays and Sundays). The Divemaster interacts with a huge green moray eel, holds a bait fish in his teeth and lets a barracuda pick it off, and you get to touch/interact with nurse sharks. When the interaction is over, you get to do a wreck dive in 35 feet of water (City of Washington and Civil War wreck).
Get the Lonely Planet book: Diving and Snorkeling the Florida Keys for a good idea of what's there. The spots I mentioned are just the tip of the iceberg.
No, I don't work for the Keys Chamber of Commerce or anything, but I have been diving and vacationing there for years. PM me if I can be of help in planning your trip. After all, "It's the Caribbean you can drive to"