they are VERY BITTER rivals. You are right in that one is based in Dxb and the other in AUH.
Overall they are both ramping up their services and both do offer similar services. However in some ways, i think Emirates has the upper hand, especially since Etihad is much younger and less experienced.
Personally, I live in Dubai but prefare to fly Etihad, as they usually have cheaper rates, with just as new aircraft and the AUH airport has a lot less walking (though the AUH airport is much older and really needs a makeover) The good news is, it is getting the makeover as I type, and a few regional destinations have started flying out of the new terminal.
Generally I wud say the food, and FA' tend to be better on Emirates, but they are not so much better that the prices deserve the premium Emirates charges.
Again, I fly Etihad 75% of the time, and Emirates 25%.