We have travelled on P&O (11 cruises), Princess (2), Celebrity(2) but this was our first on Royal Caribbean. This was also, by a fair margin, the largest ship we had been on.
Overall, I found this trip disappointing - not bad, but a long way short of what I expected. I had naively believed that as long as the prices were roughly comparable, the overall quality would be the same, although clearly one line would better better in one area than another, and vice versa. I cannot say that I think this is true of our trip on Navigator of the Seas.
The first surprising thing was that when we got on board there was no-one to give any guidance at all on where our cabins might be. One every other trip so far, there has been some sort of welcoming party/group , with people to guide you to your cabin. Here, we were left completely to our own devices. (It did not help that Royal Caribbean seems to have invented an unusually complicated way of numbering their cabins!) Now, I think this might be because of the size of the ship - the amount of time to get everyone on board pretty much the same whether you are taking 1000 or 3000 people on board, so the larger the ship, the more difficult it is to greet people as they get on. As I say, I haven't been on such a big ship of any other line, s can't be sure about that.