Just to expand a bit, Celebrity is considered the upscale sibling to Royal Caribbean. It's considered to have better food, but being pricier, generally attracts an older passenger demographic on most itineraries than Royal Caribbean.
Now I'll qualify what I just said. There are two things that will change the above. First, sailing from the greater New York area, there will be more families (who also don't want to fly) than usual on Celebrity. Plus, since I assume you'll be sailing when school is out, that will also increase the number of families with kids on the Bayonne sailings. Celebrity adds more counselors and beefs up their children's program a bit when they expect more children to sail, and I'm sure that's the case with their summer and holiday sailings.
As Cruise Fanatic said, Celebrity doesn't have ice skating rinks or rock climbing walls. If your children really enjoyed some of the more unique facilities on Royal Caribbean, that may be a good reason to consider going back on them again.
As noted, Bermuda is a wonderful destination. Lots to do, days spent out doing things, using the ship as your hotel, restaurant, and entertainment venue if you'd like. Bermuda has beautiful beaches, good shopping, and incredibly friendly and helpful people. There is a wonderful, safe and clean public bus system that can take you all over the country. You can go online and take a peek at Celebrity's shore excursions in Bermuda to get an idea of the variety of things you can do.