Oh, what a question.....
I've sailed on BOTH Carnival and Royal many times. The real issue is that things are CHANGING so fast.
My last cruises were within the last 3 months, and I found such a difference with Carnival from years past to the point I was really disappointed. Now, let me give you my 2 cents.
The Horizon and Allure are both wonderful ships and some of the best in any fleet, so you have options and will enjoy either. You didn't say how old your kids are, so I'll assume they are between 6 and 16 for purposes of my answers.
I like to sail this way: Either get a family type of suite, or get 2 rooms that are adjoining, or if the kids are older, you splurege on the verandah/balcony and give the kids an interior across the hall (hint, you can get them a connecting interior, or a next door cabin at front of ships usually on the lower decks. I haven't looked at exact Horizon or Allure deck plan prior to answering.
Carnival:
Pros: Kids love the atmosphere, the constant music and fun around the pool area. It's fun to hang out on deck, let the kids swim and you can get burgers at Guy's Burger, the Lido buffet, or the Blue Iguana tacos. The drinks are lower priced on Carnival. The comedy shows are MUCH better on Carnival. I find that the major "broadway" type shows are slightly better on Carnival. The piano bar and the promenade decks are always fun for adults. on the Horizon, there is a Family Harbor area (Get FJ, FA, FE type cabins) - some have 2 bathrooms, VERY nice as women get their own area). The Kids love this. It's also a way to grab a quick sandwich or small bite after coming in from shore excursion. The pricing is better, not just the cruse ticket, but photos, extras (like candy, pie pieces, coffees, etc.). Rooms are generally bigger (say 184 sq ft to 174) - not exact numbers, but you get the idea... I find the talent better or there is just MORE (last sailing, there were a total of 4 comedians, (to 1 on the Royal sailing, both 7 day journeys). The family suite area is very useful with xBoxes, small food, drinks, and a place to chill.. - kids hang out there and you have some peace knowing they are just down the hall
Bingo is a bigger on Carnival.
Cons: Main dining has gone downward in a bad way. The options are more limited, and the portion sizes have shrunk. They NO LONGER USE TABLECLOTHS! I hate that last bit. They are understaffed right now. May not be when you sail. But they are down to only cleaning ONCE per day (Royal does 2x). You can get more party crowds on Carnival. Last cruise there was this group of 20 somethigns, walking around with big speakers blasting rap and it screwed up my concentration at the casino even.... Carnival hides incidents (per my recent experience) of accidents and major issues and becomes difficult to deal with after initial response. (All cruise ships are basically immune under maritime laws)
They have stopped tournaments (slots, blackjack, poker) which were really popular.
Royal:
Pros: The surf simulator is a huge hit with teens (and even to 8 years old I'd say). The Allure is one of the largest cruise ships on the sea (shorter by 3 feet of the Symphony), and is a destination in of itself. My kids right now prefer the Vista (Horizon is a Vista Class), but they haven't been on the Oasis Class). The water show is awesome and you should see this. The ice skating is fun for kids that aren't from the north, or even those can get practice in... limited times during COVID, but still on the ship. The food is good, restarant and the room service has actually improved a bit. The Royal is a step higher than Carnival on the ladder, I'd say that Royal is between Carnival and Holland on expectations and "class." It's say that the spaces are nicer. The ship is quieter and the crowds are more subdued.
Cons: They are more expensive. Everything is a bit more on board, and you find that they are tapping you for small things (coffee is more, you might want to get a danish or something, that's a bit of a hit, and so on and so forth). Drinks are .25 to $2 more. Wine can be a lot more per glass that we found (note, not on the drink program). The entertainment wasnt' as good as Carnival in our family's opinion. Shows were "old" and 60 themes and other themes didn't work well with kids (or in fact even us, who grew up in the 70s and 80s).
So, what's the recommednation? My kids would say "Let's try the Allure, but we'd rather sail on Carnival." I'd say, if you are trying to go low cost, or party atmoshphere, or gambling, then Carnival is the choice. If you want to go for the ship and the meals, and you have some more money, then Royal is the way I'd lean.