Oooh, good question! Arizona is hotter for sure, especially in the summer. It has the Grand Canyon, and we don't have anything to compete with that here in New Mexico, but we do have some good hiking in the Sangre de Cristos and Rocky Mts. Arizona is more desertlike whereas we have more forest and mountain terrain especially toward the north.
We don't have any city as big as Phoenix. Our biggest city Albuquerque is roughly comparable to Tucson at half a million people. Of course, they filmed the hit t.v. series BREAKING BAD in Albuquerque and not Tucson, so that's a fun reason to visit ABQ if you're a fan of the show.
If you're into art, you'll find much more of a "pedigree" and history in New Mexico than Arizona, especially up in Santa Fe and Taos, which are famous for their art galleries and artists (Georgia O'Keefe, RC Gorman, and others).
If you like Native American culture, the Navajo Nation stretches across both states, but New Mexico also has all the Pueblo cultures and their amazing artwork such as the Hopi and their baskets and kachinas, the Zuni with their stone fetishes, and silverwork, and of course the Navajo with their famous weavings. There are Pueblo ceremonies and dancing that you can view during the summer.
Arizona is going to have more of a Anglo-European Wild West history and a Mexican-from-Mexico cultural feel. New Mexico (and parts of southern Colorado) has a unique Hispanic-Spanish-Catholic feel that has very little to do with Mexico. You can see a lot of amazing Spanish and Catholic art in New Mexico and a lot of picturesque old churches.
Also, last but not least, we have the whole history-of-the-atomic-bomb thing here in New Mexico, and Arizona has nothing like that. The Manhattan Project was helmed by J. Robert Oppenheimer at our Los Alamos National Laboratory during WWII and the first atomic bomb was tested at Trinity Site in White Sands Missile Range here in NM in 1945.
The atomic bomb history is all over the place. You can visit museums in ABQ and also White Sands itself which is a beautiful place. Almost like visiting another planet. You can even visit Trinity Site if you coordinate it (I think they only open the site twice a year and one time is in July.)