I voted Tulane. Both have similar costs (except I think NOLA's cost of living could allow Tulane to be brought a little lower, but no significant savings).
DC is a little bit of a generic major city- cool place to live, but can get the same vibe in other places. NOLA is really unique.
Both match all over, with Tulane having more going west. Tulane, I think, has a more interesting patient pop. And Tulane seems more fun/happy- though GW doesn't seem unhappy. DC has better weather.
Neither would be a bad choice. DC would be better for policy.