In terms of Dubai vs Abu Dhabi - Dubai is very glamorous; huge flashy commercial and residential buildings everywhere you look, 8-lane freeways, hotels, shopping malls. Dubai is quite small so everything is tightly packed. Sadly no character at all - apart from some signs written in arabic you really could be in any metropolitan city. Abu Dhabi somehow feels more homely - far less showy, more calm, and more real. It's more of a sprawling city and you can see huge empty sand-filled lots everywhere.
Sheikh Zayed Grand Mosque - Abu Dhabi
If there is only one thing you see in the UAE, make sure it's the Sheikh Zayed Grand Mosque. It is truly breath-takingly spectacular and absolutely free. You should know their dress code is strict. The website says
I thought I would be ok in ankle length pants and a short-sleeved t-shirt and scarf. But you actually need to be covered down to your wrists, so I was ushered into the basement to line up for an abaya which you just slip on top of what you're wearing (yes SUPER HOT). It may have been way too big for me but being in a mosque I actually found it comfortable. I would have felt out of place if I was wearing my normal clothes.
Also a huge aquarium inside the mall. Because why not.