I think the big variable here is what you will do with the resources.
I'm about to graduate from a school in the northeast like American. The classroom experience is not substantially different from that at any good school of our size. It could even be worse, because our professors are hired solely for their research and not for their teaching. When they turn out to be excellent instructors, it is a pleasant bonus for the university.
Given that the classroom experience is nothing special (very good, just not something impossible to replicate elsewhere), what is special? Smaller classes, access to professors, library resources, student support, opportunities for individual and group research, resources to pursue strange and outlandish projects, exceptional arts and culture programs accessible by the entire student body (music, art, dance, theatre), enormous student-run initiatives, and more that I cannot name. All of this comes at a tuition price, whether your family pays it or scholarships pay it.
Some of my friends soak this up. Their education is rounded and their social and non-academic time is engaging and sometimes connects back to their academic work, and there is, here, a culture of learning that can be pervasive outside of the classroom, if you buy into it and dive into it. Some of my friends live in the libraries--not a bad choice, our libraries are very good. I also have some friends who could not be more disinterested by all of this, and yet they get the same tuition bills we do.
Point: private schools are expensive but offer a lot that large state schools cannot; the price tag might be worth it, but IS NOT worth it if you won't take advantage of the resources available--otherwise you're just paying more money to do the same stuff you could at a state school.
Two of my good friends just graduated from large state schools (one of them ASU, actually, so in your neck of the woods). They had very large classes, professors were difficult to engage with, and there was not nearly as nice a ratio of opportunities on campus per students (with the exception of drinking). But they also paid next to nothing.
Point: public schools are often cheaper (much cheaper if it is your state's public school) and offer a great deal, but you are competing with an enormous student body for resources, access, and opportunities. If you can do that, then that is great and you'll succeed marvelously. If you can't or aren't as comfortable doing that, it's a lot easier to slip through the cracks and just float through school. Then again, you're floating through school for free, so maybe it isn't as bad as it would be for 5K a year.
You can get a great education at either American or at Arizona. At Arizona you will need to work harder to make it your education and not just coast along; there will be fewer resources to help you along your way. Given American's size and proximity to the playing field of your interests, it might be much easier. But it will be cost more and take you farther from home. Grades, test scores (GRE, LSAT), recommendations from professors, research, internships, work experiences, etc. will all matter when pursuing work beyond college in your area of interest, and you can do well in all of these at either school. And you might also change your interests.
The big question I would think about is, if you can get a great education at either school, which school do you think would be best suited to the way you want to pursue that education and to which school you think you are best suited?
Good luck.