What side had an advantage in professionalism of the army?

The South had a significant advantage early in the war due to the high fraction of experienced senior officers from the South who chose to side with their native states. Almost 1/3 of the military officers of the US Army resigned their commissions to join the South and the percentage was even higher among the more senior officers of the Army. It could be argued that most of the best commanders in the US Army were from the South. Even the most capable Northern generals were only considered average compared to the brilliance of some of those commanding the Confederate forces. Note that the President of the Confederacy, Jefferson Davis was a graduate of the US Military Academy. Early in the war a higher fraction of the Confederate rank and file soldiers had previous military experience than the Union forces. These advantages were a major factor in the many early successes of the Confederate forces.

As the war continued, Northern commanders gained more experience and improved in skill while the South lost many of those highly skilled military leaders as battlefield casualties. By the end of the war, due to attrition, the advantage had shifted in favor of the Union forces; although the South probably still had the better top commanders, the North had significantly more soldiers and the experience gap in the enlisted men had been closed.