The capital of the Confederacy was 110 miles from Washington D.C., which got me thinking whether it would've even made any difference if one side would've conquered and occupied the other side's capitol. It seems like you had "border states" throughout the war, and I guess both sides just agreed to march through them and ignore their affiliation.
But all this begs the question: what were the specific objectives of each side. Did the Union see the complete destruction of the Confederate army as the only way to end the war? And were the Confederates trying to destroy the Union army too, or just outlast them to the point where the Union called off hostilities?
The affiliation of the border states was of vital political and strategic importance to both sides throughout the war. Lincoln for instance insisted that his generals prioritize the "liberation" of Eastern Tennessee from the Confederacy, since most of its districts had voted against secession, even though his generals advised against it and ultimately the attempts failed. Missouri, Kentucky, and Maryland also had regiments recruited from their populations by each side. Confederate strategy also included plans to "flip" Kentucky (Bragg 1862 and 1863), which had they succeeded could have won the war by threatening/cutting Union traffic along the vital Ohio River.
The objective of the Union was to re-unite the country by suppressing the "rebellion" within the Southern states. That meant destroying or dispersing its armies and seizing and occupying the majority of its territory. All while keeping the CSA diplomatically isolated and un-recognized as a sovereign nation by other foreign powers, most notably Great Britain.
The Confederacy just needed to get the Union to give up fighting and have its sovereignty/independence recognized by other nations. It didn't really need to seize territory or destroy Union armies, which meant the CSA could afford to trade territory for time. Confederate strategy did include threats and attempts to "invade" Union territory as a means to undermine popular support for the war within the North as well as the claims by the Union government that the CSA was illegitimate. For the CSA capturing Washington D.C. probably would have been a winning coup, demonstrating that their army was powerful enough to guarantee protection for their territorial claims. They would not have sought nor found it necessary to occupy D.C. for a prolonged period of time for this effect to be achieved. Unfortunately for the CSA, Washington D.C. was almost permanently occupied by an army as large as any the South could bring against it, in addition to being protected by the Army of the Potomac in the field.
At the beginning of the war, the strategic objectives the Union needed to achieve were far more complex and difficult than those needed by the CSA for victory. That is why so many in the South were so confident they would win in the end. Many Southern leaders drew comparisons to Napoleon's failed attempt to invade Russia in 1812 as a precedent for the impossibility of subduing a country as large as the CSA was in 1861 (i.e. comparable to the size of Western Russia). In the end that confidence was misplaced.