Well, horses for courses I reckon.
Housing: DXB offers a greater variety with a split between villas (a.k.a. houses) and apartments. Villas more expensive to rent in AUH but apartments cheaper in AUH. DXB also more modern, with most buildings being new(+-5 years).
Cost of living: DXB more expensive to live than AUH, in general.
Traffic/Transport: A no contest here. AUH!!! The only downside to the small size of AUH is the parking problem. "What do you do if you see a spaceman..... you park your car man." You can drive from one side of the island to the other in about twenty minutes. A taxi ride will cost you no more than 15 Dirhams.
Education: Someone else might assist on this one. I know everything already!
Seriously though, I don't have children so can't speak about this.
Abu Dhabi has a much slower pace to it. If clubbing and partying and life in the fast lane is your scene then stick to DXB. For a more relaxed lifestyle go for AUH.
Abu Dhabi airport is more accessible than DXB for the above reason and the drive there is easier.
A bit criptic in description but won't bore you with details for now.