i think royal and disney are more similar than different. for the record, although i love wdw and enjoy my disney cruises, royal is my favorite cruise line, so just keep that in mind when you read my comments.
(i've also been on celebrity and norwegian, but neither floats my boat, so to speak, so we've sailed exclusively royal and disney for quite a few years).
before dynamic dining, i would have said rotational dining on disney was the biggest difference, but now that royal has "classic" dynamic dining which is basically the same thing, that's no longer true. i think the MDR food on royal is better, but disney's buffet and palo are better than royal's buffet and specialty restaurants (though there's really not too much of a difference). on quantum, cafe 270 was fantastic and my favorite lunch spot on any ship ever. royal also has more food available at night - part of their buffet on quantum/anthem is 24-hours, there's a cafe on the promenade open all night (it may close for an hour or two for cleaning at like 3 a.m.) and there's a pizza place open til 2 a.m.
i guess that's a good segue into the nightlife on royal, which is why i prefer it. i'm a night person, and disney is boring at night, quite frankly. royal has shows and activities until 2 a.m. or later, there is always at least one pool and hot tub open (unlike disney, which closes them all at i think 11 pm), and places to snack at night.
royal has a casino, which disney doesn't. i dont gamble, so not a huge difference for me, but definitely a difference in general.
royal has an adults-only area called the solarium with a couple of small pools, hot tubs, and a bar. it's beautiful, but the pools are so small they're not really usable as pools. i'd give royal the edge for atmosphere, but disney the edge for the adult pool itself.
no fish extenders on royal. i love them on disney, so that bums me out. most royal cruises have a "roll call" on cruise critic and sponsor activities through there, though, including a "meet and mingle" which royal itself sets up for people on the roll call - they have snacks and drinks and meetings with the officers, and raffles where you can win prizes. roll call members commonly organize other stuff - on quantum we had a gift exchange (didnt make up for no FE, but it was something), slot pull (which i did even though i'm not normally a gambler - everyone contributes and all the $$$ is put in one machine and everyone takes a few spins, and you split the winnings/losings), cabin crawl (see everyone's cabins), and pub crawl (where we got a special of 5 drinks for $10 or so plus free snacks). not all roll calls do all that, though - it just depends on whether someone is willing to organize.
both disney and royal have similar plays/shows, with comedians, musicians, etc. i think disney's shows, like golden mickeys, villains tonight, etc, are cute. but i think royal's are better - some of them are very close to broadway-quality. apparently the cast of "we will rock you" on anthem was in the london version, so serious actors. royal does not have a dedicated movie theater like disney, though sometimes they show movies in the main theater, including 3d movies.
we've always lucked out with good service on royal, but i think overall disney's service is probably better. we've never come across an unhelpful cast member on a
disney cruise.
i don't know which anthem itinerary you're looking at, but i've found labadee to be equivalent to castaway cay. there is no adults-only beach on labadee, though.
i'm sure i will think of more the minute i post this.
but hopefully this'll give you an idea, and if you have particular questions, just let me know.
We went on quantum, anthem's sister ship, this past january, and absolutely loved it. it's a different experience than disney, but still great. i've been on 7 royal cruises, so if you have questions, ask away!
