We have cruised on both. They offer very different experiences. Disney is by far the most family friendly cruise line in the industry. It provides entertainment geared toward both adults and children, and most can be enjoyed by both together. Carnival is not nearly as family oriented. Although Carnival offers a very good kids camp, it still feels like an add-on. To give you a feel for the different priorities between the two cruise lines, the amount of floor space dedicated to children on Disney ships is taken up by casinos on Carnival.
Not sure of the make up of your family and your interest. If you only have adults & teenagers, then Carnival will be perfectly fine. If you have children under ten, Carnival will be ok, but a Disney will be terrific. However, if anyone in your family is looking for a little slot or card table actions, you can scratch Disney off your list. There is no casino aboard a Disney ship.