@FlemGem: You do realize Lee could not put anyone in command of the western theater, right? He was in command of the Army of Northern Virginia. It was President Davis that chose who commanded each theater of war. And in any case, until the battle of Shiloh, Albert Sidney Johnston was the commander in the west, and he outranked not only Jackson but also (by seniority) Lee. Then they put Joe Johnston in command out west, who was also senior to Lee.
That being said, I vote Lee all the way. Lee was brilliant tactically and strategically. He made mistakes, yes, but he is human. Lee realized what he was up against. He realized he would be outnumbered at every engagement, and would probably face multiple invasions of Virginia at a time. He maneuvered across the state, staving off Union incursions and delivering truly stunning victories, both through his own direction and playing to the strengths of his subordinates. The two ways the South could have won the war would have been to a) make the North sick of the cost in lives and resources and quit, or b) win a victory on Northern soil and demoralize the North, possibly gaining foreign recognition and aid in the process. He invaded the North twice, and came close to winning twice. When it was clear that he no longer had the resources to mount another invasion, he conducted a defensive campaign against a superior foe and held out longer that nearly any other army could.
Yes, Lee owes credit to his subordinates and his men. But he was the one that was able to coordinate them all around his plans, and give his subordinates enough leeway to do what they did best without giving them so much that the army degenerated into a mob. Lee's soldiers would have done anything for him; many were still not ready to give up when he surrendered at Appomattox.
Grant was one of the better Union generals, but far from their best in my opinion. Reynolds and Hancock were both good tacticians and would probably have made better army commanders than even Grant. Reynolds was offered the job and refused before the Gettysburg campaign (where he was killed), and had Burnside listened to Hancock the entire Fredericksburg debacle could have been avoided. Grant's tactics consisted of using the advantages he had: more men and supplies. He pushed and pushed and pushed until Lee could not physically resist anymore. In the process, we ger battles like Cold Harbor, which make the massacre at Fredericksburg look like a pillow fight. Yet Fredericksburg ruined Burnside's career, and Cold Harbor is but a rarely-mentioned footnote on Grant's.
Given an even fight, Lee wins. Given an uneven fight, Lee still wins. Given a lot of uneven fights that Grant can recover from and Lee cannot, and Grant wins. Barely.