Choose the school, not the battalion.
They're both excellent schools but quite different in terms of their strengths and the surrounding environment. Georgetown is going to have a much heavier emphasis on international relations and anything related to politics. For STEM subjects Vanderbilt has the edge.
As to the social environment, Vanderbilt / Nashville will be more balanced ie more intellectually/politically diverse than Georgetown. Vanderbilt's student body is a bit more diverse academically as well: it comprises not just STEM and liberal arts but also a top-notch School of Education and an excellent School of Music.
It was a long time ago, but I was born and raised in Nashville and graduated from George Washington University. My high school was literally on Vanderbilt Campus and both my parents and sister went to Vandy.
@thibaud basically has it right, but I would add a few things:
Each school Is the academic tip top of its cohort of schools—Vandy in the SEC and G’town in the big East.
Both schools recruit nationally and actively pursue a diverse student body. However, Vandy is still a decidedly Southern School and G’town, an Eastern school. G’town is a Jesuit school and Vandy has a very proud Divinity School which feeds mostly into the mainline Protestant Churches.
If internships interest you, particularly if related to military/national security, then DC is your place. In DC you can’t swing a dead cat without hitting someone you know, love or hate, from the news. Many of them are on University faculties when their party is out of the White House. Imagine C-Span as your classroom. My freshman year Poli Sci professor was a Retired career Foreign Service Officer and former acting Ambassador to India. He was an unofficial advisor to Tip O’Neil, long time House Speaker. The class met at 6:15 pm and he inevitably arrived half lit, having just left “Social hour” at the Speaker’s office. We knew that whatever they talked about would probably be on the front page of the Washington Post the next morning. His former student was my boss when I was an intern in Lloyd Benson’s office.
In Nashville you can’t swing a dead cat without hitting someone deeply involved in music. My son was a handy violinist and fiddler. When we visited family in Nashville, I would take him to a jam session on Sunday night’s at a place called the Station Inn. It was open to everyone and I mean everyone—from my 12 year old to someone who looked like he/she just fell off a Greyhound Bus. At one of these sessions, Jerry Douglas, the world’s foremost Dobro player, walked in and joined. No one batted an eye. Nashville is also a world class business center, so internships abound there as well.
If you like college football weekends, Vandy is in the SEC and G’town is in the Patriot League (FCS). ‘Nuf said.
We have known a number of kids from here in Nebraska to go both places and never heard a discouraging word. Both our DS’s applied to and got into Vandy, but neither chose it, much to my chagrin.
Congratulations on earning these opportunities.