It's unlikely that there will be warm relations between the two, but economic realities will mean that there will be a lot of trade between them; like Canada, it's right next door, and it has a long open border. A lot of this depends on just how the south achieved it's independence. If it was because of British/French intervention, then the Union just might get it's revenge when/if those two are distracted by WW1.... or it might just sit on the sidelines and threaten the CSA into doing the same. If the south did it totally on their own, then the USA will either:
take a good look at their army and at those of Europe, make a lot of upgrades, and go back into the south for revenge, or
let them go, and in time, accept that the south is independent.
It's hard to say just which they would do; smart money would bet on the latter, since war weariness would be dragging down the north...