The fact is that slavery as an institution was dead in the West within a decade or two at most of when the Civil War occurred.
So, let's say that the South had peacefully seceded in 1861.
I predict that every major European power would have stopped trading with the CSA within a decade or so due to public pressure against slavery. Once that occurred, essentially the entire raison d'etre for the CSA would have evaporated. The South would be forced to end slavery as an institution or face complete economic collapse.
Furthermore, both the North and the South would have found themselves disadvantaged economically due to the split. Northern textile manufacturing would be at a disadvantage due to increased costs to access Southern cotton, and the South would be at a disadvantage due to increased costs to access finished manufactured goods.
With slavery dead and the CSA's reason for being eliminated and a number of disadvantages to both countries becoming ever more obvious and a realization in the North and the South that both countries were weaker on the world stage, the pressure to reintegrate the would have been enormous and ultimately irresistible.
In the end, the CSA would have ultimately rejoined the Union within a decade or two at most.
Some things would have definitely been different. Over 600,000 Americans would not have died. Untold millions of property damage would not have occurred, leaving both sides more wealthy. There would have been no Reconstruction Era.
The only question in my mind is how the Civil Rights era of the mid 20th century would have gone. It is certain that racial discrimination would have been rampant in the South in this alternate reality (it was rampant in the North as well, although less obvious due to the demographic differences). However, in a world where the South had successfully left the Union for a time once already, would Northern liberals have been in as powerful of a position to force change in the South as was the case in the history we know? Probably not. I think the Civil Rights era would have occurred ultimately, though probably delayed by a decade or two, possibly occurring in the 1980s rather than the 1960s.