UNC has a better rank than AU in terms of Political Science, but will American offer me better oppurtunities?
American is in Washington, DC, and the other is not. You will have plenty of opportunities at American.
You can’t consider the benefits without also considering the cost. If cost is a concern and you get in-state tuition at UNC, then that tips the scale fully in UNC’s favor. If not, then look to AU.
I’d be wary about rankings since they are so dependent upon criteria, as I’ve seen some with AU near the top too. I’d say the proximity to DC gives AU a big advantage, but that advantage is up to the student to leverage it. After visiting the DC area, it is amazing to now know how many resources are at the fingertips of students (Metro is easy to use too). These include internships, speakers, think-tanks, conferences, museums, as well as all the US government. Most likely, your professors will have contacts throughout Washington as well. So for opportunities, I’d give a big nod to American.