Just so we have context, is it because you aren't bothered by having a casino on board (this is common) or that you aren't bothered by cigarette smoke (this is less common)?
So I feel like your colleagues have engaged in a bit of hyperbole. I don't believe that there is anything on Royal that "blows away" Disney. It's probably more fair to say that Royal and Disney are pretty close in terms of the metrics cruisers care about. However, there are a few areas where I feel Royal does hold a decided advantage - price, activities, number of shows and the opportunity to spend extra money.
Price - this is obvious. Disney cruises costs more than some luxury cruises, yet their competition is the family-friendly brands like Royal and Carnival. This is not necessarily a huge sticking point.
Activities - Disney has the usual cruise activities, plus the Midship Detective Agency game and Aquaduck (on some). Royal has that, plus rock walls, flow rider, ice skating, and a couple of other activities on Oasis/Quantum class ships.
Extra Money - A lesser price brings more ways to separate you from your money (via your SeaPass card). Royal has more specialty restaurants, drink packages and places to shop. And those shops are in places designed for you to see them, whereas there is a good chance that you may never encounter the gift store on a
DCL ship.
Coffee - DCL's coffee is complete undrinkable sludge. I'm someone who practically has a cup in hand at all times, and I had to go without for a week. Royal's coffee has a decided advantage in that a) it's brewed, and b) it tastes slightly better than gas station coffee. Yes, this is a backhanded compliment.
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Now, there are some notable differences, which are good or bad depending on your opinion:
Specialty restaurants - there are generally more of these, and they allow kids to eat there.
Royal Promenade - This does feel like a stretch of indoor mall at times, but I cannot tell you how much I missed not having this when I was on Fantasy and it was raining on a sea day.
Dining - No rotational dining. I don't feel like this is a big deal either way, but you may feel different.
Kids Clubs - The structure of Oceaneers vs Adventure Ocean are noticeably different. Oceaneers has activities, but they're structured to be more encompassing across a larger age group. Adventure Ocean's programs are more specific in terms of ages, although this may change if there are less kids in the program. My kids have enjoyed both, and more importantly, they appreciate the things each program brings to the table.
Pools - Oasis class has more pools, and they are not overrun with kids
People - There are several Royal ships that have a higher passenger capacity than DCL Fantasy. I haven't been on anything smaller than the Oasis class, so I can't speak to the crowding there. However, I can say that I felt more crowded on the Fantasy than I did on an Oasis class ship. It seems contrary to what the numbers might tell you, I know, but it might have been that the percentage of kids was lower. OTOH, having about 2000 more people unloaded onto a private beach makes for a more crowded experience there.
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There are some areas where I think things are roughly the same:
Cabin service - I can't say that I've had a cabin steward that was more exceptional or much worse than any other. I do think that DCL does a better job, but again it's probably a few little things here and there.
Dinner service - Service can vary quite a bit depending on who you get. I would say that the service on my last two cruises (Fantasy and Harmony) was the best I've had, and that the servers absolutely earned the extra gratuities we gave them.
Food - Like service, it can vary a lot. The best food I've had was on the (old) Regal Princess in 2006-2007, and that was due in part to a dessert chef who made some amazing souffles. Every. Single. Night. But overall I can't say that food has ever been so bad that it was inedible.
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And then there are some things that Disney does better
Private Beach - I've not been to Coco Cay, but
Castaway Cay is nicer than Labadee, if only because the main beach is less rocky. My opinion here might be tempered with the fact that there were more people on the beach at Labadee than there were at Castaway Cay.
Quality of shows - No surprise here, but Disney's theatre shows are quite good. Mind you, Royal's aqua shows are also very good, but those are exclusive to the Oasis class, and aren't enough to tip the tables in their favour, IMO.
Characters - Yes, Royal has Dreamworks characters. But I'm not a big fan of the non-speaking characters. When you see the princesses and how they interact with guests, that's what impresses me.
I might add on more. That's probably quite a bit for now.