I would go to Duke. I visited Vandy, and liked it a lot, but it just won't open the same doors that Duke will. In terms of job opportunities, the Duke degree will give you more than the Vandy degree. Both do very well in the Southeast (with the edge going to Duke), but if you want to go to NYC, SF, LA, DC, Boston, or anywhere else outside the Southeast, Duke is clearly better.
Also, Duke's JD/LLM program seems very interesting and is a great opportunity. Sure, you'll pay a bit more to go to Duke, but I think the improved job opportunities and earning power the Duke degree will give you makes it a good investment. You'll obviously do fine going to either school, but I personally would go with Duke. Either way, congratulations on the Duke acceptance and good luck!