If the CSA gains independence because of foreign intervention or has it forced upon them then there will be bad blood and a desire for a rematch, probably around WW1 with the likely Confederate benefactors tied up in Europe
If it's more peaceful, North America may well end up knowing the same peace it knows in the OTL. A big issue would be territory, what does the CSA get and what does the USA stand to lose, does it get Kentucky, Maryland, the Indian Territory, the New Mexico Territory, does Missouri get split into a Confederate State of South Missouri and a Federal State of North Missouri?
In time it might be that the CSA becomes an ally of the USA and they enjoy a respectful partnership, slavery will be an issue but realistically the CSA could not maintain slavery in an increasingly industrialized world. Unless owners start making slaves work in factories which will piss off an awful lot of white Confederates. One option could be a Confederate alliance with the Empire of Brazil, the only other big slaveowning nation though it's got an abolitionist Emperor. Combined a CSA-Imperial Brazillian Pact could result in a more modernized and especially industrialized Brazil (to offset the much more agrarian CS reliance on the industrial USA) which would make the power-balance in the Americas much more interesting
And without a border there's less chance of the US helping Juarez against Maximillian I so the Empire of Mexico may end up lasting and becoming the feeble third sibling of the Richmond-Rio-Mexico City Axis