Well, let's do some reasonings here:
The Union will have made some progress in tearing down the Confederacy. That said, there are several points of contention between the two countries:
-Where to draw the borders (West Virginia? Tennessee? Transit rights on the Mississippi?)
-Tensions over slavery (Southern Bounty Hunters and Northern Abolitionists)
-Unionist Minorities in the South
The Confederacy will amaze everyone by spontaneously giving up slavery the day after independence and building a libertarian utopia that annexes every nation down to Argentina in a new, egalitarian future.
OK, now that I have everyone's attention...
Slavery is the ideological foundation of the Confederacy; this is as important as Sharia Law to the Taliban or the "inherent superiority of the Yamato people" to Japan. It can't be compromised or lost without basically giving up on everything accomplished in its banner.
This is 1/3rd of their people, and they have to suffer abuses like having their children sold for money.
What happens if they get a chance to do something? Would they simply be content for their freedom, or would they seek revenge? Now, how many guns and soldiers does it take to keep a third of a nation's population profitably enslaved?
Snake suggests that the Confederate Army winds up running the show. I suspect that internal security would be a hugely important, if not all important task. I therefore think that the fugitive slave hunters of the 1850s will be codified and empowered as an arm of the Confederate Government. And if that task comes to dealing with "Uppity" free blacks or "Poor Cracker" farmers, I suspect this police force will probably handle the job.
I'm really unsure that the CSA would turn into a military dictatorship; I don't think he army will ever really surpass the political will of the plantations, and I think said plantations will probably look to build their own version of the Pinkertons.
I see a completely plutocratic South, one that has enshrined slavery evolving into one that has enshrined no rights for its lower classes. State's rights be damned, the planters will build a strong state that overrides the states if that serves their interests.
The United States will continue to move forward as the Confederacy makes its moves towards a police state. As in the Antebellum South, abolitionists are dangerous revolutionaries and need to be silenced. People that try to educate slaves are fomenting nascent rebellions, and god have mercy on any socialist who sets foot south of the Mason-Dixon line.
I'm also unsure that the Confederacy will be interested in massive expansion. The United States will guarantee Mexico's independence and will be unenthusiastic of any other moves south. The Confederacy could potentially aim for Cuba, but that leaves it with the ominous position of having another class of colored people to abuse.
The Confederacy must, above all else, avoid a rematch with the Union. Thus enjoined, the CSA would have to know either to remain neutral in global politics or pick the same one as the United States.
All of this is possible, and the CSA could survive as a quiet, dictatorial state. It would be a dark, cold place filled with violence, secret police agents and armed borders, but it has the slavery so desperately craved by its leaders intact.
Otherwise, the South explodes in a red revolution as the poor and the slaves make common cause and overthrow the planters and their secret police, which probably draws some kind of US response.