How do these schools compare in their international relations/studies programs? Also what are the pros and cons of each school? I only want to apply to one of these.
They are both very strong in IR. I would say the difference comes down to the campus structure and amenities. American provides a more “traditional” college experience with one campus with a quad, dining hall, etc. GW is much more urban with no defined campus, it’s split among to locations with the main downtown area and the removed Mount Vernon campus, less traditional dining, etc. Most folks have a preference for one set-up over the other.
They are both located in a wonderful location for those with your interests. But for American, the School of International Service is one of the largest of its type.
It seems that you’re interested in Washington D.C primarily. I’ve heard that American University is more of a “traditional” university with a small campus. Georgetown is a lot more open.
both have highly ranked IR programs at both the undergrad and graduate level. but having said that…GW is clearly superior in every conceivable way. assuming you can get into both, the only rational reason for anyone to go to american over GW is if the former offered you more financial aid money (which is an unlikely scenario given GW’s considerably larger financial resources and aid programs). otherwise, i honestly can’t fathom why anyone would want to go to american. it’s much more out of the way, which makes getting to internships more of a hassle (plus their neighborhood isn’t nearly as nice as foggy bottom). and frankly, GW is somewhat more well known than american, particularly outside of the immediate DC bubble
^Not surprisingly, the above post is from a current GW student. Biased much?
American is in a great neighborhood, with more of a campus, less urban feel. Students at AU have no problems getting and getting to internships.
Both are solid choices for IR. I recommend visiting both and seeing which clicks with you.
My visiting options are minimal if there are any at all since I currently live overseas 
It’s worth considering that GW has a much larger endowment than American, which would undoubtedly allow their students some more opportunities. However, I think it’s really a matter of personal preferences regarding campus life and how much you value a “traditional” college experience. They’re both solid IR schools. I think GW is a little more well-regarded (http://foreignpolicy.com/2015/02/03/top-twenty-five-schools-international-relations/ ), but they’re both great.
@jdam9828 Are you familiar with Google Maps’ “street view?” If so, I suggest checking out both campuses virtually using Google Maps. I’ve written a long post with several links to locations on GW’s campus but haven’t been able to find it for you. However, here are a few offhand:
Kogan Plaza https://www.google.com/maps/@38.8995778,-77.0476945,3a,75y,211.08h,88.6t/data=!3m6!1e1!3m4!1s5GiQLl3R3lmp1IHLiaUrFQ!2e0!7i13312!8i6656 This is kinda the “center” of campus. Kogan is an outdoor hang-out and is right next to Gelman Library and across the street from the Marvin Center. Also, the construction you see is where the new “Superdorm” is located.
Science and Engineering Hall https://www.google.com/maps/@38.9006965,-77.0489483,3a,75y,218.5h,95.7t/data=!3m6!1e1!3m4!1sGLjG4-BIcCKKvCbQ2EFnWw!2e0!7i13312!8i6656 This is the new science building, which is huuuuge. It’s at another corner of campus and is near the Metro, GW Hospital, GW Admin offices, as well as Whole Foods.
Elliot School https://www.google.com/maps/@38.8959863,-77.0443132,3a,75y,21.67h,98.81t/data=!3m6!1e1!3m4!1ssTFsYU7MBTMYFCmJ8D7Elw!2e0!7i13312!8i6656 If you’re going to GW for IA, this will be your home. It’s at the far southern end of campus, but there are quite a few dorms in this area as well as being across the street from the State Department.
American:
Bender Arena https://www.google.com/maps/@38.9376802,-77.0888502,3a,75y,217.54h,85.98t/data=!3m6!1e1!3m4!1sP9NvBJ8_H2VUtnEmFilb2w!2e0!7i13312!8i6656 Sports center, there’s food nearby, and it’s kinda central to campus.
School of International Service https://www.google.com/maps/@38.9361036,-77.0884738,3a,75y,173.71h,92.37t/data=!3m6!1e1!3m4!1sYnWfGVTB-9UJg8EQ9sWTvQ!2e0!7i13312!8i6656 I’m assuming that this is where their IA classes are, but I could be wrong.
Either way, here are some links to give you some places to branch out from. Once in a location, you can click and drag to look around or click on the chevrons on the road to navigate. I know this might be a little tricky, but it’s a great way to see an area. There’s a HUGE difference in feel between the two campuses, and if you can’t visit in person, at least look at some maps/pictures.
Good luck!
I went to GW law school, so I can say that I did not like that GW really didn’t have a campus feel because they are pretty much part of Wash DC. But I gotta say that GW law school was the best school to work while going to school.