Okay, the most obvious difference would be when Disney is having a special like Kids Sail Free. Those are just about sold out right now, so the Disney price is from some quotes I did a few weeks ago.
These are for 7 night Mexican Riviera cruises, 2 adults and 2 kids out of Los Angeles right around March 4th.
Disney - For two adults and two kids in a verandah room, their quote was right at $2650, including all taxes and port fees.
Carnival - Generally the cheapest so we'll start there. They're super close. It would be $2580 for a verandah room. This is a case where adding in the "nickeling and diming" makes Disney the same or less. Two soda cards (and if it was my family, we would want 4) would run them about $100 taking that cruise up to $2680.
Princess - The price with taxes and port fees would be $3088.36 up front.
Holland America - That would be $3286.16.
Royal Caribbean - No sail dates out of Mexico so we can't compare that one.
That's just an example. As I said, often times Disney is definitely more.
It's a little hard to compare right now, because unless they run a special, the cheapest time to book Disney is right when it comes out. The opposite is often true of other cruise lines. Disney's prices go up over time because their ships fill up. Other cruise lines offer more last minute deals because their ships aren't always filling up.
I have several people who are booked on 7 night cruises in the summer of 2012 on Disney. One is a family of 5 that booked the morning the cruises were released. Their ocean view cabin was $1281 per person for the first two adults. (additional people in the room are less) You'll see that's a good deal less than Royal Caribbeans price of $1680 per person that you found and that cabin that sleeps 5 is much bigger.
There are deals to be found on Disney, you just have to know when to book, and where to look if you missed that!