As a person that knows horses and is very familiar with riding, one of the advantages that the south had over the north was that southerners usually had their own mounts, the federal army provided the horses for the men. Rider/horse trust and understanding takes a long time to be archived. I wonder if besides that fact and maybe better officers from the start, the southern cavalry had any other " advantage" over the north, like tactics, weapons etc?
Early on in the war (so 1861 and most of 1862), the Union cavalry lagged behind the Confederate cav. on pretty much every quantifiable way. There were a number of reasons for this, the most obvious being that the CSA represented a less industrialized portion of the country, and hence many of the fighting-age men had at least some passing experience with riding and horse upkeep (for lack of a better term). And while Union soldiers hailing from Illinois, Ohio, or even Pennsylvania certainly had experience on horseback, the more agrarian south had a jump-start on their northern counterparts when it came to effective riders available at the start of the war.
There was also the difference in theory between north and south in terms of how cavalry should be used. In the north, cavalry were thought of (primarily) as a compliment to the infantry. By this, I mean they looked at cavalry as a component of a larger military apparatus rather than a separate, independent force capable of achieving its own objectives. Thus, early on, the Union cavalry primarily guarded the flanks, screened infantry movement, and protected against enemy cavalry crashing through the flanks. Thus, while the southerners perfected their raiding methods and behind-the-lines incursions during those early months/years, the Union had its cavalry handcuffed (which hampered their effectiveness when they were finally let off the leash).
The Confederates, on the other hand, had more confidence in their cavalry and commanders to operate with relative discretion and broad objectives. From the beginning of the war, they moved independently (they weren't anchored to the infantry's movements), made raids behind the lines, and harassed larger Union bodies to keep them from massing or gathering their full strength against smaller Confederate forces. JEB Stuart, N.B. Forrest, John Mosby: all of them were absolute terrors in the saddle, and they operated with a lot of discretion to achieve objectives with only broad instructions and orders.
When the Union cavalry went up against Confederate cavalry in 1861 & 1862, it was usually a drubbing that saw the Federals routed during pretty much every engagement. The Union cavalry just wasn't as good on horseback, which was largely the result of green troops with less time in the saddle, officers with murky objectives (and little latitude to operate independently and on the fly), and Confederate opponents who knew the region better and often had the support of locals.
There was also the intimidation factor: early on, the Confederates were simply more capable in the saddle, and they knew it (and made sure their opponents knew it)! Yet as time went on, the Union began to catch up, and fast. By 1863 up to the end of the war, the Union weeded out many of the bad cavalry officers, and promoted the good ones. This then trickled down, and produced better troopers who were given better training at riding, but even more importantly, caring for their mounts. It's remarkable, really, that in just under 2 years, the Union was able to field several cavalry detachments that were the equal of the Confederates.
As far as advantages, there really weren't a lot that the Confederates had early on aside from experience, daring, and confidence (which did/does count for a lot).
[Sources: James McPherson, 'Battle Cry of Freedom'; Bruce Catton, 'Mr. Lincoln's Army' & 'Glory Road']