Take a look at RCL website. We are on the Allure but I've looked at teh differences. Quantum/Anthem have extras such as skydiving simultor, SeaPlex with bumper cars, roller blading, North star observation thing for great views, and Dynamic Dining (TBD when/if it gets implemented on Oasis/Allure).
Both have rock climbing, FlowRiders, great kids club, DreamWorks stuff, Broadway shows (but different shows), etc
I'm not sure if Quantum has same number of neighbourhoods. Someone else may be able to clarify. (It doesn't look like they offer cabins such as Central Park and Boardwalk).