Atlantis is a great hotel for children but its quite expensive for all the incidentals, so be aware. An alternative is the Hilton on Jumeriah Beach - older hotel, great location, fab swimming pools etc.
For Dubai - would suggest an abra on the river, heading to the museum, the beach, Wild Wadi water park.
For Abu Dhabi - the corniche is nice to look at and the beach has been really improved, Sheikh Zayed Grand Mosque, Ferrari World, F1 track - the go karting is great, but its way to hot to be doing that at the moment.
You could do an organised trip from either place into the desert - some off roading and then a meal, belly dancing, camel rides etc. Also a boat/fishing trip.
Not sure when you are planning on coming, but August is Ramadan, so no food/drinking allowed between sun up and sun set, and then the end of August will be Eid, so everywhere will be very busy. about 8 years ago
Dubai is an awesome destination-summer or winter.Would highly recommend the Wild Wadi,Dubai Dolpinarium,Abra Ride,The Desert Safari,Dubai Musuem -all in Dubai and The Ferrari World in Abu Dhabi. My 10 year old son has visited these places and vouches for it. Not to miss Burj al Arab,Jumeirah Beach and of course,Burj Khalifa. Great Deals available in the Malls 4 u too.
Save on hotel costs-try 3 star hotels-in the Riga Area...can highly recommend Versailles Hotel. about 8 years ago
Atlantis and its Aqua Park are the best for a family in my opinion, although I agree that it won't come cheap, it's totally worth it.. Not the most comfortable rooms but it will do... Try Zabeel Saray if you're looking for new comfortable rooms, and it's a 2 minute drive to Atlantis, and 5 to Wild Wadi about 8 years ago