Emirates Adventures has a pretty good evening desert safari out of Abu Dhabi (http://www.eatours.ae/tours.asp). How long to you have in each place? Dubai has all the mega malls and worlds tallest building. The gold and spice souk are good to walk through even if it's touristy. I wouldn't really buy anything there, and if you do make sure you have your bargining skills tuned in. From there you can take an Abra across the Dubai creek and there is a tourist souk. Again, good for people watching. The Walk @ JBR is my favorite in Dubai. Nice to sit outside along the beach and have lunch or coffee.

As for Abu Dhabi, there is the Grand Mosque. Have lunch or coffee at one of the restaurants along the Corniche and beach. Of course there is the Emirates Palace that is worth taking a look at (and the really do sprinkle gold on your dessert). Right now there is construction going on, so best to take a cab as I'm not sure if you can walk to the front gate yet or not. There is kayaking through the mangroves with a chance to see a flamingo. If you are in Abu Dhabi on a Friday, then Friday brunch is a must. Every hotel has a brunch. Although not the largest selection in town, but the Viceory Yas Hotel on Yas Island makes it unique as you can look out over the Formula 1 track. If you're luck they might be doing time trials or classes that day. If you are a racing fan they even have tours of the track and behind the scenes.

Hope this helps.

You could try the regular desert safari but another really fun tour is a Falcon training one where a guide teaches you how to train a falcon. it's pretty exciting. im not sure if all tour agencies do it but i know one Royal Shaheen Events does this. I remember having a blast when we filmed with them. Also, i would recommend a cultural heritage tour by SMCCU (Sheikh Mohammed Centre for Cultural Understanding) where you'll get a visit to the Grand Mosque in Dubai and try out traditional outfits, followed by a "bastikiya tour" of the heritage site and ends with a traditional emirati meal and coffee with the guide where u can ask him as many questions you like about the culture and even misconceptions about the religion. it's pretty great!
I would also suggest you end your day by going down to the Creek (which is near by) and take the Dhow (A wooden arabic boat) across the creek for just AED 1 (thats less than half a dollar!). Make sure u don't use the ones that charge you AED 100 or AED 20. it's the same boat ride that you get.
Enjoy!