First of all, Atlanta was a major railroad hub for the South, a distribution point for manufactured goods heading west and south, and agricultural goods headed east and north. As the South started the war with a mere fraction of the nation’s miles of railroads, Richmond and Atlanta were the keys to denying its effective use to the Confederacy. The southern economy was already feeling the protracted pain of the Union naval blockade, the Atlanta railroad hub was really the heart of what remained of southern trade and commerce by 1864. With its destruction, it would greatly hinder the ability of the South to move troops and supplies between states, further hampering efforts at cooperation between rebel states.
Secondly, Atlanta would represent a symbolic victory. To capture it in 1864 would be to capture the first major southern city with the exception of New Orleans, and deep in the heart of the Confederacy. Capturing it would be an important nail in the coffin of the South, basically making it only a mater of time before the war had to end in Union victory. Southerners knew this, as did their President Jefferson Davis and their commander Robert E. Lee. Every southern soldier, upon hearing of Atlanta’s capture, must have known in his heart of hearts that, despite whatever bravery and fortitude they put forward in the coming months, they had lost the war, plain and simple.
Thirdly, Atlanta was, along with Richmond, the largest manufacturing center in the South. Much of the war materiel the CSA depended on in order to continue waging war came from Atlanta's factories. Factories that had become doubly important since imports of war supplies were reduced to a trickle by the blockade.
Lastly, and this cannot be overestimated, capturing Atlanta at that point in 1864 was such a major public relations victory, such an overwhelming piece of good news and progress, that it was a major factor in Abraham Lincoln’s re-election as President. If the city had not fallen, if Grant had not conquered most of Virginia by that point, then we would have been inaugurating President McClellan in early 1865, who had vowed to make peace with the South. That was their last chance for victory: making the war so long, costly and bloody that the northern public was no longer willing to pay the price. They would elect a peace candidate, and the South would win independence by default. Atlanta’s loss was the beginning of the last chapter, the final days, and the end of the Confederate States of America.