Celebrity are RCI's premium brand so it makes sense to expect a better standard of service, otherwise they'd be shooting themselves in the foot. Independence is aimed directly at the 'family market', so school holiday periods are going to be hell if you don't want kids around you. If your looking at RCI as an option, the Radiance Class ships tend to be a lot less child friendly & are suited more to the 'traditional' cruise experience.......Wilba .
RCI ships tend to be very impressive but you should expect lower standards of food, service and space. They are more family orientated and less formal. On the other hand the fare should be lower. Personally I do find RCI good value for money.
Have you thought of Cunard? Celebrity are almost an American version of Cunard (Cunard having an Anglo-American feel).