I can't speak to the comparative quality of each weapon.

The Union used the Springfield rifle because that's the rifle made for the US Army by the federal arsenals, such as the one in Springfield MA (where the gun gets it's name from). Roughly 60K were made of the 1855 model, which was used at the beginning of the war. As these were stocked into federal armories, when those armories were seized by Confederate troops, those arms were used by the Confederacy. However, the vast majority of the arsenals to make more rifles were in the North. Meanwhile, in the North, the rapidly expanding Army of the United States (the volunteer army, as opposed to the United States Army) needed more rifles then were stockpiled. Both ended up buying and licensing the Enfield rifle, though the North stopped as manufacturing of the 1861 Springfield rifle had increased enough that the use of Enfield rifles declined in Federal armies. The Confederacy however never managed to grow a suitably large enough manufacturing base to supply itself with it's own rifles. It would continue to (try) and import weapons until it's surrender.

The North for the most part had the ability to supply a uniform rifle to it's armies. The South was more reliant on whatever they could get. Some southern troops used springfields, some enfields, some other weapons. As to quality, I don't think both would have switched, even if one was better then the other. The north was already producing large numbers of springfields, and those were doing the job, and it kept the supply line simpler (one reasons the army didn't switch to breechloading and magazined rifles, despite the advantages), and the South was so desperate for guns that they'd use anything, despite low quality.