First off, congratulations!! Those are both great schools and they are both offering you very nice financial packages! They are both in a great location and you will have many opportunities at both. GW is better known and has a better reputation generally, but really, they’re both great schools. Your success at school will depend much more on your personal drive and what you make of the experience than the reputation of the particular school. I’m a firm believer in following your heart – but a well-informed heart! 
To me the starting point is your statement that your heart has already chosen AU. I’m glad to hear that you’re not leaning toward AU solely because of the money. You mention the research scholars program, the double major, and the amazing people. The fact that the money is better – $14 k per year better – only helps to solidify that choice.
So, the question is, should you choose the school that you are not in love with, where you won’t be a “research scholar” and where you won’t be able to do a dual major, where you’ll have to pay $14 k per year more, so that you’ll have the intangible benefits of a school with a somewhat better reputation.
In my opinion, unless someone can give some clear evidence that you’ll be better off with a GW degree than an AU degree, you should follow your heart.
Here’s my story, which may or may not be helpful. I followed my heart to choose University of New Hampshire (for undergrad) over Colby College. Colby was always one of my top choices, and UNH was my “safety,” but I just fell in love with it after visiting and my heart just said UNH even though Colby had a much better reputation. I was really impressed by the friendly and down-to-earth people I met there, and I guess one other factor was that I craved a larger school after going to a small high school. Anyway, I ended up very happy and successful at UNH, and it was able to get me into my “dream” law school.
Upon graduation from UNH, I wanted to go to law school in D.C., and I applied to only two schools – American University and Georgetown University. I heard back first from American, and they gave me a generous merit scholarship, and I was excited about the school. (I was really impressed by everyone I met and they told me they would support a research effort I wanted to engage in.) Then I received my acceptance from Georgetown, with no money (just some loans). Part of me felt I should be loyal to AU, which was so supportive and generous, but my heart told me to go to Georgetown, despite the higher cost. I was so, so happy with that decision, because I loved Georgetown and it was very clear that Georgetown law students had significant advantages over American law students when it came time to search for a legal job.
So, it seems I’m offering you a mixed message. Not really. Keep in mind (a) in my case, my gut and heart said Georgetown, not AU, and I went with that; (b) when I was going to school, tuition was much cheaper than today, and my parents could afford to pay the cost differential (so I wasn’t terribly tempted to go against my heart for financial reasons); (c ) in the field of law, your law school makes a HUGE difference in the kind of job offers (and salaries) you can receive, especially the all-important first jobs, so I KNEW that the Georgetown degree would pay off in the long run.
If you’re still wondering if a GW degree would somehow serve you much better in the future, you could think about what type of job you’re hoping for upon graduation, and then make a few calls to people who work in that field. (I think most people would welcome a short and polite call from a student.) You could ask them if they would give more attention or preference to an applicant with a GW degree versus an AU degree. The caution there is that you could end up going in a very different direction career-wise. In my case, I KNEW that I wanted to be a physicist when I applied to college! And I ended up graduating with a dual major with Political Science and International Affairs and a minor in Russian. Go figure!
So maybe you don’t want to over-think it, and just go with your heart!
Good luck. You will not go wrong! They are both great schools!
Go to American!