We've done Royal (Odyssey of the Seas) and Disney Dream. Atmosphere is the largest thing that was different. Now, I'm not saying one is right/wrong--but you will know if this is for you or not. My kids tend to be more the nerdy, academic variety. They are not into the latest greatest brands, they are not exceptionally sport minded kids (think cheerleaders and football players more than bowling club, JROTC type of "sports"). My kids (all teens and young adults) felt like the kids on board were the "popular" kids and they didn't fit. On Disney, they made friends with other cruisers very easily. For the adults, nice--but a bit more plastic surgery, women wearing string bikinis, etc. than maybe you find on the Disney. It is just a different vibe--maybe it was also because we cruised over spring break season too (with Royal)?
We are a Disney family. We don't want to get all dressed up to go to dinner (Royal technically has relaxed the formal dining and dress up dining each night at dinner--but I assure you it is still very much a "thing" there). We want to go to dinner in sundress and sandals and a polo and nice shorts or something like that. We felt VERY underdressed each and every night on Royal in the main dining room. Don't get me wrong, I promise we didn't dress like trash (think a Tommy Bahama shirt and khakis, my kids wore collared shirts and nice shorts, my daughter and I wore summer dresses)...but we were out of place and servers even commented to us "don't worry, we allow the casual dress now"...I never felt this way on Disney--it is relaxed and chill.
entertainment shows--they were equal. Royal did not disappoint here.
customer service--equal, Royal staff was as good as Disney. Not better, just equal.
Royal was exceptionally crowded, even in the adult only areas. Plus we didn't like the solariums are like huge indoor pools with echos--we loved the adult areas on Disney that are outdoor and relaxing and you hear the ocean.