Plenty of cases out west where union infantry had inferior weapons compared to the csa.
Grant rearmed many of his regiments with Enfields after the capture of Vicksburg.
One difference is that the union generally raised new regiments, instead of reinforcing existing regiments.
Wisconsin was an exception to this.
The Union also relied heavily on European immigrants, many of whom had no experience with firearms (and marksmanship training was not optimal). .
Plenty of Americans had no experience with firearms. And if they did it was properly shotguns and similar.
There is no evidence that the average soldier knew anything about marksmanship. And with a few exceptions they never learned anything about during their service.
In comparison plenty of immigrants had served in European armies.
Most of "German"armies was based on universal male conscription, including Prussia.
So many of the "Germans" actually had military experience. The same in Scandinavia where a large part of the young men served in the army during their early 20ties.
Some Germans had combat experience from the rebellion in 1848.
The south did have one advantage when the war broke out. The military academies in the south gave them a good number of regimental level officers who knew what they where doing.
They also started out mobilizing about a month earlier then the union.
If there was any geographic difference I think it was more east / west then north south.
When we are talking infantry that is.
When looking at cavalry I think the south was clearly superior during the early war.
And in artillery the union had a huge advantage with the regular army batteries making up about 1/3 of the artillery.
(based on the oob at Gettysburg)