Obviously the defense of the capital, and it’s being a constant objective for the Confederate Army’s dictated where many of the great battles were fought. Would the Union have been less aggressive if defense of Washington was not paramount. How deeply would into the North would the war extend? Would the psychology of the Citizens of the North been less protective of maintaining the Union, if it did not mean the capital would sit so close to the border?
As a note: DC isn't (or wasn't) in the heart of Dixie. It is south of the Mason-Dixon line, but located on the border of Maryland and Virginia (Land was taken from Virginia and Maryland, but the Virginia portion was returned in 1857, over a decade prior to the Civil War). Maryland, though a slave state, ended up not seceding. So DC wasn't in the heart of Dixie: it was on Dixie's frontier. That's not to say that the positioning of the capitol wasn't helpful or harmful to either side's efforts, but potentially a helpful refocusing of the question.