I think everyone is giving you good information on the differences. I have stayed at the Hyatt and at the Tamarijn and I liked both, although they are completely different. I have young kids also and kids staying/eating for free was a big draw at the Divi/Tam. We also loved being right on the beach. My wife likes to sleep in (meaning past 6 am in my house) and I like to rise early with the kids. We would simply slip outside early in the morning and go swimming right after waking up - which was fabulous. The beaches at the Divi/Tam are beautiful and we took advantage of the free snorkel gear, use of a sunfish sailboat, kayaks, floats and noodles, etc. The restaurants were OK but in my opinion were not exceptional which was a bit of a downer given all the great restaurants on the island. It is easy to take a bus or taxi to the downtown area or other areas of the island. My kids loved riding the golf cart shuttle around the resort, including going across the street to the Divi Links which has a great water slide at its pool. Another thing to take advantage of is free entertainment each night at both the Divi and the Tam. My kids loved the Carnival Show, the Mariachi Band, and they even had clowns to amuse the kids one night. We don't spend much time in the room but you will probably find the rooms lacking at the Divi compared to the Hyatt. They are small, dated, sometimes the grout around the tub is not in great shape, and our television barely worked.
I think you will have to forget all of your Hyatt expectations and go with the flow at the Divi. The views, water and beach are incredible. The kids will love eating at the pool bar which has pizza, chicken nuggets, fries, onion rings, panini sandwiches, etc. They can also get ice cream at each pool bar. Let me know if you have any more specific questions.