It kind of depends on the ages of your kids and what they like to do.
Here is a link to the activities for both ships:
http://www.carnival.com/onboardexp_home.aspx
http://www.royalcaribbean.com/findacruise/experiencetypes/experiencetype/home.do;jsessionid=0000ac7IgucMWkNAVHlYm8V_Vn_:10ktdmm42?br=R&exTypeCode=8
In my experience I prefer Royal Caribbean over Carnival. The service is much better, and it is a lot cleaner. However, food on Carnival is less expensive, the food is somewhat better and rooms are a little larger. The shows on RCL are a little better, especially for kids, Carnival is more Las Vegas style shows. Carnival is known as the party boat, which in my opinion isn't always good for kids. On RCL you have a lot more choices as to what you want to do for entertainment. Also with Carnival, you may pay a lower cost up front, but often times when you get on board you will find a lot things aren't included and you may end up spending more.
I suggest looking over the activities for both cruise lines and pick the one you think suits your kids best. You know aht they like, so it will be easier for you to decide. Both are pretty nice cruises, it also depends what boat you are taking. In my opinion, Royal Caribbean is a better cruise line, Carnival is more budget friendly and has more activities for kids.
Princess and Holland America are both well known for their Alaska cruises. However, these may not be as age appropriate for your kids, it tends to be older couples.
If you think that your kids would like the Activities on Royal Caribbean I would try that line. Overall, I have had much better experience with them over Carnival. But your kids will probably find more to do on Carnival. Again, they are both great ships, but it just depends on what you think your kids will enjoy most.
Hope this helps!