How did the Union and Confederacy compare in terms of resources, leadership, and military strategies in the period 1861-1863? Why was that important to the outcome of the war?

Both the North and the South during the Civil War had specific advantages and disadvantages that would contribute to the eventual outcome of the war.
One advantage the Union had over the Confederacy was that the Union had a powerful central form of government that could create and enact laws to be implemented by the entire Union; this included taxes. The Confederacy had a weak central form of government with very limited powers under Confederate president Jefferson Davis. The process of taxation was left in the hands of individual southern states, which, seeking approval from their constituents, implemented very few—if any—taxes. So we see the Union had tangible assets to back their paper money and a powerful central form of government with the ability to levy and collect taxes, while the Confederacy had nothing to back their paper money with, leaving the South with relatively little buying power with foreign nations to purchase supplies and goods for their troops and an inability to levy taxes on the states because of weak central form of government.

The Union had a definite edge in terms of resources at the outset of the war. It had more food, industrial capacity, and people. It also had greater gold supplies with which to pay soldiers and buy weapons overseas. The North also had a navy with which to attack Southern ports. The South had a smaller population, and many members of this population were African slaves the South would not even consider arming until 1865. The South had farmland, but a lot of it was cultivated for cotton. The South lacked major gold supplies to pay soldiers or gain credit overseas, which was a moot point because the Confederacy lacked international recognition from Britain and France.  
The South had an advantage in military leadership. Robert E. Lee and Stonewall Jackson could not attack their home state of Virginia and stayed with the Confederacy. The Army of the Potomac had to endure the likes of George McClellan, who was overly cautious, and Ambrose Burnside, who was inept as a leader. It was only in the West that Northern generalship excelled under Ulysses Grant in the Tennessee and Vicksburg campaigns. The North did have Lincoln, who would eventually prove himself to be a better president than the South's Jefferson Davis, although he was hated by many in the North from 1861-1863.  
The South had the better strategy, at least initially. Its goal was to wear out the North and to fight a defensive war. This would have been the best strategy, as the Army of Northern Virginia was repulsed both times on Union soil in Antietam in 1862 and Gettysburg in 1863. The North, under Winfield Scott, had the better long-term strategy in the Anaconda Plan, however, which was to be a blockade of all Southern ports and for the North to control the Mississippi River. In 1863, the Union finally took over the Mississippi River with the fall of Vicksburg, and the ports were all closed with the fall of Wilmington, North Carolina in 1865. After 1863, Southern privations started to show themselves in the civilian population and on the battlefield, and more Southerners argued for peace.