My first non-
Disney cruise was on Adventure of the Seas. I knew it was going to be Disney, and I was prepared for that, so had kind of low expectations. We were WOW'ed! The ship is lovely, and we found most things to be at least as good as Disney, except for the main stage shows. We were on a TransAtlantic cruise, so there were some additional entertainers, some of whom we had seen on Disney.
The ship is nicely laid out, with the Royal Promenade in the middle, which gives you a nice place to people watch indoors (in case the weather is bad), which you just don't get on Disney. The main dining room food was about the same, we liked the Windjammer better than any of the Disney buffets since it's open most of the day, and had a bigger variety. It doesn't have the outdoor eating area.
The rooms are a little smaller, and unless you're in a suite there's no bathtub, but we found that there was a bit more storage for all of our stuff.
We're an older couple not traveling with kids, and we had a great time. We're taking our third Royal cruise in 13 days, and our fourth in 17 days! We haven't taken a Disney cruise a while, and probably won't until there are either new itineraries or better prices. We've found that Royal is just a better deal for the money, and we haven't felt that we sacrificed anything.
If you have any specific questions, don't hesitate to ask!
Beth