I am a Marriott Koolina and Aulani owner that happens to be a local homeowner living 5 minutes away. Koolina is a resort community that has hotels, timeshares and homes in the resort area.
There is a lot to do on Oahu as there is hiking to the lighthouse at Makapuu, Pearl Harbor memorial with USS Missouri, Punchbowl, Polynesian cultural center, museums, sugar plantation, etc. You can make it as adventurous or relaxing as you like. Some like to spend their whole vacation and never leave the resort except to venture out for food. Others go out every day and explore the island. Then there are those that choose to spend some time at the resort and go out and explore the island. Only you know what type of family you fit into.
If you are the park commando type and staying at least 8 nights in Hawaii, I would do 4 nights on each island. That will give you 4 days on each island if this is your once in a lifetime trip to Hawaii. If you do island hop you will probably lose most of that day. If you have breakfast and checkout at 10am, load vehicle and drive to airport for interisland flight by 1200pm. Flight takes all of 30 minutes. Deplaning, picking up your luggage and then getting a rental car will take you to 2pm or so. Then driving to your new hotel will put you right at check in time. It will be at least 5 to 6 hours of travel time to get to your new hotel on another island. You could stay as others have suggested 2 nights at Marriott Koolina then do 2 nights at Aulani.
If you are the more relax at resort or the hit the parks in the morning then go back to resort for nap/pool time then I would say staying on one island is more your style. You could do 4 nights at Marriott and then 4 nights at Aulani to see the difference of both resorts and that would give you enough time to experience the resort. I would not like to pack, goto airport, go to new hotel and then unpack. We like to check into one resort for our whole trip at WDW.
Differences between Marriott and Aulani has been mentioned previously by others. Marriott is quieter and laid back vibe compared to the louder and more park like setting of Aulani. You can do 1br at Marriott. When you arrive ask for a roll away and you can put that in the study by the master bedroom. Gives you older one a little more privacy and he can have their own "room". The sofa bed in living room converts into a queen sized sleeper. At Aulani your kids will all be in the living room and one will have the murphy bed under the TV and the others will share the full size sleeper sofa. Bathrooms are about the same between the two. Both have full kitchens and washer/dryer in room. Aulani provides the same kitchen ware as the DVC units at WDW but you also get a rice cooker at Aulani. I find that the Marriotts at Koolina and Florida to provide a more complete set of kitchenware compared to DVC. Marriott Koolina 1br villas are 920 square feet vs Aulani 1br villas at 756 square feet.
As others have posted, Aulani is more "busy" and Marriott is more quiet. I think your young ones would like Aulani more as there are a lot more things for them to do and with a lot of Disney touches makes it magical. Marriott is a great resort but is a more laid back vibe. They both provide resort amenities and feel. As cmrdgrs has said the pool hours at Aulani are shorter than at Marriott. The adults only infinity hot tub is great at Aulani (especially at sunset) but the Marriott has 7 hot tubs and do not close till 11pm. Also the temps at Aulani are not as hot as the Marriott. Aulani has a great lazy river and two slides that are great. One is a long dark slide and the other slide you use a tube to ride down that ends in the lazy river(make sure you do the double tube with your husband or children). Marriott has one dark slide that does not compare to the Aulani one. Aulani has many crafts and activities that Marriott does not do. Also there is the trail that you use an Ipad to unlock some of that Disney magic at Aulani.
We like both resorts a lot but spend more time at Marriott now. When my son was younger he did not like to stay at Marriott Koolina and preferred Aulani. When he was around 12 years old, he told me he prefers to stay at Marriott instead. Now we only stay at Aulani when we have visitors or when I take my nieces and nephews for staycation. We have celebrated every Christmas at Aulani but this year, will be the first since Aulani opened, that we will be at Marriott instead. Now he is of that age that Disney is not magical anymore and the rides are to tame. We were in Orlando in June and he was sure happy that we did not go to WDW since it was closed. Will be back in Oct for Halloween and told him we will be doing WDW especially ROR. He was not to thrilled and asked to skip WDW like the summer trip. Told him that he will love ROR like he does Space Mtn and Expedition Everest.
PM me if you have any questions and I will help you as much as I can. If you can tell me what your idea of a vacation in Hawaii is, that would help me figure out what you would like to do. Many folks have said that this is their one trip to Hawaii but have now have visited many times. Whatever you decide to do, I am sure you will have a great stay.
