Summary:
Please note that this is for the Anthem only. Each ship and each class of ships has different features. What we had on Anthem won't exist on the Serenade probably because its an older ship. Shows, food, restaurants, etc are all different on each class of ships.
Check in - Disney. It was a quick check in on RCL but the waiting and unfriendly staff put DCL ahead. I see fellow passenger had a different experience later in the day so maybe this was dependent on time of day.
Lunch/Buffet - tie. Both have banquet quality food. I don't think that either one was better than the other. Again, we aren't foodies but we found the food to be comparable.
Ice Cream - Disney. RCL had ice cream but it always seemed to need a refill and you only ever got 3 choices, chocolate, vanilla and strawberry. And some of the days, Vanilla wasn't even an option. But almost every time we went to the station, one of them needed to be refilled and it was beeping for at least 45 min. Its also not open for as long as Disney is either.
Shopping - Disney. This is personal preference but we don't care about the fancy shops. Some do probably. But I prefer more the regular people kind of shopping vs expensive Jewelry/watches, etc. It was also strange to walk into The Shop (RCL labeled items) and see an entire section, about the size of the clothes and merchandise, was alcohol and cigarettes. I know they sell both on DCL but its in a separate shop away from the regular merchandise and I prefer it that way. The two times I bought something in The Shop I was behind someone buying alcohol and pissed that they couldn't just open it right then and start drinking it. They didn't' realize they had to wait til they got off the ship to drink it. I've never had that problem on DCL or have to explain to my 8 yo why those men were so upset.
Shows - Disney. Again personal preference but when you are traveling with a kid, you want them to enjoy it too. Since the only show she could go to was Spectra's and it was based before her time, this ship was not kid friendly as far as shows were concerned. Now, we take note that some class ships might be better than others but this ship in particular was not our cup of tea.
Indoor Pool - Royal. Obviously because Disney doesn't have one. That was a huge point for us since the weather was so dreary.
Overall food choices - edge to Royal. This comes with a ** because you have to pay for those choices. BUT you paid less for the cruise overall so you can afford to throw in a couple of paid for food items.
Drink Choices - Royal. With the freestyle machines (again, I don't know if those are on all the ships or just Quantum class) that makes the drink choices much larger for soda drinkers. We both got the Coke cups and used them quite a bit. We did enjoy having more choices than coke, diet coke and sprite on DCL. But again, this was something you had to pay for. I had wondered if you could get water easily outside of the WJ and the answer is yes. They had a water machine with ice in the Cafe Promenade.
Information/Apps - Disney....just slightly. We found most crew members on RCL to be annoyed by our presence. They would give us information when asked or would help when needed but it was rarely done with a smile on their face and they most certainly didn't care if you existed or not. Many times the Compass didn't give the full picture of where something was or for how long it would take. And the app was nice in that you could make reservations and see your account balance, it didn't give locations on the app of where things were sometimes. So I feel like the DCL app was a hair better and the Navigator is better than the Compass.
Nightlight - Royal. Again this may be ship specific but the light in the bathroom had one small section that would stay on when you turned the light out to serve as a small nightlight when the bathroom was otherwise dark. I appreciated that for nighttime bathroom trips.
Storage in the room - Anthem. The layout of the room was more thought out than any Disney ship I've been on.
Room Key/Wowbands - Slight to Royal. You don't get your room key at check in, they are outside your room when you get on the ship. A part of me didn't like this because who's to say that some creeper wouldn't take one of the keys and was waiting in your room for you. Supposedly only one key was outside the room but in our case, all 3 were out there. So my advice is that if one of the keys is missing...be on guard walking in. BUT we really liked the wowbands. Similar to the magic bands but thinner and you couldn't choose a fancy color like magic bands, we still liked them. It was nice to have since we needed them for almost everything we did.
Room noise - Slight to DCL. Only because we could sometimes hear our above neighbor walking. Never had that problem on any of the 4 DCL ships.
Ship Decor - Anthem. I went specific to Anthem because again, other ships may not be this pretty. I still love the classic style of the Dream and Fantasy (the true comparison to Anthem) but Anthem was a very pretty ship.
Shower - Anthem. I enjoyed not having a shower curtain cling to me.
3rd bed - DCL. Now, I've read reports that our setup may not have been the norm (see Happilyeverafter's report above) but for us, the winner was DCL.
Activities-Adult: Tie
Activities, Kid: Tie. Anthem had roller skating, bumper cars, indoor pool but DCL has characters, scavenger hunts, crafting activities,etc.
Pizza: Royal
Internet: DCL. Personal preference here. We get the smallest package on DCL. Just enough to make sure our house sitter/pet sitter to our 13 year old westie is doing ok. Not much surfing. So yes, Voom was faster but it was double the price. So I go with DCL because we don't need it to be exceptional.
Pool Towels: DCL. I didn't like checking the towels in and out...but the wowbands came in handy for that.
Smoking areas: DCL. See my rant above.
Bathroom cleaning: DCL. Although my bathroom was clean for the most part, the toilet bowl brush cleaner was left in the bathroom. I didn't like that. I have never seen the cleaning supplies on our 4 DCL cruises. We also had to ask for more toiletries and I've never had to do that on DCL. Even in the later cruises when they stopped giving you a new set each day, when they saw we were low, more was left. I had to ask for new ones this cruise.
Outlets: Slight to RCL. The Dream and Fantasy have a better outlet situation than the classics do but RCL had slightly more outlets and even USB ports which was a nice touch. Again, probably specific to the newer RCL ships.
So that's it for me! I hope someone finds this useful and be sure to ask questions if you have any!