Both Wyoming and Arizona are fantastic trips!!! We have been on both trips and loved them both with our two DD's who were around 10 and 13 at the time. They both include many outdoor activities and native culture--especially in Arizona with the chance to experience some native american culture. One of the primary differences is the temperature. If you are going, Arizona and Utah will be very HOT!!! If you have trouble with being outdoors in the heat, Wyoming may be a better choice. It actually snowed in August when we were in Wyoming. While Arizona/Utah has amazing views of the Grand Canyon, the badlands, red rocks, arches, etc. . . Red Cliff Lodge was a real treat. It was nice to stay put and to be able to sign up for various activities--the ATV trip through the canyons was outstanding. That being said, I think the scenery in Yellowstone along with Brooks Lake Lodge in Wyoming can't be beat. We loved staying there and having the lodge to ourselves (our
ABD group that is). The lodge has an amazing spa for mom, a herd of horses for DD's, fly fishing for DH, and fantastic food and entertainment for all the family.
Good luck making your decision! I don't think you can go wrong with either trip.