I've lived in both places. Edge goes to Nashville. I don't like Gallatin though -- I'd chose one of the other suburbs.
D.C. has a very high cost of living (you would be borderline poverty if you lived by yourself in the District) whereas Nashville is relatively inexpensive. Also, people are nicer in Nashville and it is less of a "rat race." You can venture off to the mountains or parks on the weekends and not feel stuck in the city as you would in D.C.
Also, if you love sports, Nashville might be better too. We have the Titans, Predators, and many other amateur teams. You can find plenty of sports action here. D.C. will have lots as well. But, you have to deal with ridiculous traffic and difficulty attending games.
D.C. is good if you like big city living, more restaurants, and all the political atmosphere that comes along with DC. Also, DC residents tend to network and get a better chance at government-type jobs. I got tired of DC, the weather (cold winters), the crowd, and the expense ($2k a month for a small 1 bedroom). Also, it is known for high taxes, including an income tax (TN, by contrast, has no state income taxes). Check out how much it costs just to register your car (by the way, ditch the car if you move there).