Have done both, now more loyal to RC. However, compared to when we last did Disney in 2013 here is what I recalled (in no particular order);
1. Disney is definitely pricier if you are happy not getting a big suite on RCL. Many
DCL fans argue that RC will nickel and dime you for non alcoholic drinks etc., some experiences like the Escape room, and that is likely true. However you would need to balance this against the extra costs for the cruise itself.
2. I found that on the largest of RC ships, the crew felt a bit overwhelmed with the volume of passengers. Less so on 'smaller' ships like Quantum class or Freedom Class. Having said that that was on my last experience on Harmony when it was a new ship and judging by the workers' comments I sense that there were a lot of kinks that have to be sorted out. I was last on Anthem and I did not sense any of this stress at all.
3. Palo on DCL was probably the best cruise restaurant I've eaten at. I liked it better than Remy, as well as when compared to Chops, 150 Central Park on RCL. But that was back in 2013 and I hear that it may not as be as high quality as before.
4. For young kids (i.e. 12 and under) can't beat the offerings that DCL has to offer. Now they have more themed offerings like Avengers etc. I hear. However my kids in their teens now say RCL has much more to offer: flow rider, laser tag, the trampooline thing, ifly. The new virtual reality thing on Odyssey looks cool but haven't tried it.
5. As adults we found the larger RC ships had more places to just sit (that were not out in the sun) chill compared to Disney. From what I recall from the Disney Dream and Fantasy, adults could sit in the cafe, and in the main atrium area, but there weren't a ton of seats that didn't require you to be on the pool deck. RCL ships have the promenade in its larger boats (or the Esplanade on Quantum class), central park on Oasis class etc. - seemed to be more places to relax indoors when you get sick of the sun (first world problem I know!)
6. I'm not big into the stage shows, but Disney has its name associated with it which provides some legitimacy production wise. Having said that I've seen some of the broadway musicals on Royal - while I'm not a big musical, the production value and the talent seemed top notch to me.
7. I personally like the decor on DCL as well as the newer RC ships (harmony, Symphony, Anthem and later). Some of the eariler RC ships have decor that is a bit outdated but this is a very very very minor point in my eyes.
8. If you have kids who want to see characters, DCL is perfect. However see to what extent they are still offering this these days. RC used to have the Dreamworks characters available for photos etc.
9. DCL cabins have the split bathroom idea to allow someone to use the shower/bath separate from the toilet. Makes a difference with more people in the room. Getting two cabins on RCL is do-able price wise, and not far off from a DCL booking (when I last checked - haven't checked recently).
10. Found the dining in DCL main dining rooms a bit better than RCL's, but both in the end serve food that is a ton better than what we would cook ourselves so can't go wrong with either.
Hope that helps the OP.