My background: 3 RCCL cruises between 1998-2008. Twice as adults only, and once with DS4. Left from Miami twice, and Galveston once. Been on Viking, Enchantment, and Rhapsody. Have done both EC and WC itineraries. Just returned from
DCL Fantasy out of Port Canaveral, WC itinerary C. All cruises have been 7 days. Never stayed in concierge. I have never been on Oasis or its sister ships though. If RCCL has dropped in quality since I've been, then my assessment doesn't make sense. Corrections - 1998-2010 on Voyager (2) and Enchantment. My mind is slipping as I get older
Conclusion: DCL with kids. RCCL adults only
Why:
MDR - RCCL is fancy/more formal/quieter and has good ocean views for the entire dining room. I don't know if they rotate to different ones now or not. DCL is nice and fun but more oriented toward entertaining children. Food and service are great on both ships.
Buffets/QS - pretty much equal between the two. DCL inclues sodas, and there are beverage stations open 24/7 that have sodas, water, tea, and hot drinks.
Midnight buffet - Great on RCCL. DCL has a really week, mainly dessert buffet, on Pirate Night. It has very little artistic design, and the food isn't anything special.
Pools/Hot Tubs - RCCL's pools are bigger, and they have more hot tubs. The adult area has way less kids, is more serene, and has sun protection. DCL pools are more geared to kids and smaller. They only had 2-3 hot tubs, and at least one, if not 2 of those, were in the adult only area.
Physical Activities - RCCL seems to have more, because they have ice skating, rock climbing, the flow rider, and basketball. DCL has the Aquaduck. Both have mini-golf and ping pong.
Gym - RCCL is bigger and usually has a hot tub in it. DCL had plenty of cardio equipment and some weight machines, but really no room for functional fitness (boot camp, free weights, calisthenics).
Spa - RCCL has jacuzzi and sauna access when you book an appointment. DCL charges extra for the Rainforest Room, but it seems to have more stuff. I didn't actually do it.
Ports/Excursions - Pretty equal between the two. DCL excursions meet on the ship, which makes tendering go very smoothly. RCCL always seemed more stressful when tendering. Grand Cayman is a tender port. DCL has towels ready available. They also have towel return, water/lemonade/tea, and hand sanitizer when returning.
Private Island - RCCL goes to Labadee, Haiti. It's very safe, but there are armed guards at the border, kind of far from the main activity though. It might make you feel a little guilty. They are both nice and have a lot of nice beach area and stuff to do. DCL has
Castaway Cay, which seemed to have more pay excursions. I don't remember many at Labadee, mainly just beach, water activities, and snorkeling.
Cleanliness - Both are very clean. However, DCL is very pushy, in a good way, about getting people to use hand sanitizer before entering the MDRs and the buffets. They use the wipes and have people stationed at every entrance handing you one. RCCL seemed to place dispensers at the buffet entrance and squirt people at dinner when entering MDR.
Food Allergies - DCL seems a little neurotic about it if you tell them, which could be good or bad.
Ship - RCCL ships feel more open, and you can see the sea from all over the ship. DCL seems to hide the sea more. You couldn't see it in most of the bars, and the windows in the some of the MDRs are not very big. Animator's Palette has no windows. You could feel like you were in a hotel on DCL. This is a personal preference thing. I like seeing the sea.
Decks - RCCL seems to have more areas where you can sit on deck and look at the sea and read. It was hard for me to find anywhere to sit on DCL, because everyone was reserving seats with their property. Hardly anyone was actually in the seats though.
Shows - RCCL is more adult. DCL is more kid friendly, but still enjoyable my most adults. DCL movies can't be beat, and the theater is 3D. They had very recent movies running constantly during the day all week.
Bars - RCCL bars are usually more open to the common areas, so it's easier to people watch. Most of the DCL bars are in enclosed rooms without windows.
Cost - RCCL seems to be lower priced to me, but that could be because we get a loyalty discount. Plus, they give a Texas resident discount in Galveston.
Specialty Dining - I have not done this on RCCL. However, I don't think anything can be Remy on DCL. It was the best meal I have ever had, and we have eaten at some really well rated restaurants in New York and San Francisco. Palo on DCL may compare more to RCCL, but we didn't do Palo.
Cabins - DCL mostly has split bathrooms, where the small tub/shower and a sink are in one room, and the toilet and a sink are in the other. That was nice with 3 people but still convenient with two. RCCL used to put the bed near the verandah. DCL puts the couch there.
Luggage - DCL luggage delivery was very timely. If you do onboard airline checkin, you can leave your bags out the last night and pick them up at your home airport. No lugging stuff through customs or the airport.
Either way, you'll have an excellent time and won't question your choice of doing a cruise on either.
Thanks! 
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