It depends. The Soviets would probably try to keep relations neutral and help establish organizations early on since the South would have more proletarians. As for the government, it'd probably be socially conservative (and slavery would of ended somewhere after 1880) but also have populist roots. Southern politicians were pretty populist when it came to Cuba, which they viewed as a great opportunity to help expand the Southern economy and make it more independent. Assuming history goes this way (Pierce keeps on encouraging filibusters and such to take Cuba and after a while Cuba falls) then let's review:
1917-1924: A divided US would of either meant that fewer US troops would get involved in the Russian Civil War, or that the British would encourage the CSA to also send troops, making it about equal to real life. Soviet foreign policy towards CSA is roughly same as USA, probably a more determined effort to establish organizations. (See above)
1924-1939: Same until Great Depression. Huey Long would probably be around with his populist views. Communist Party CSA possibly accuses Long of encouraging fascist tendencies (as CPUSA did IRL) or gets caught up in populism and endorses him under the banner of progress. Regardless, Long would probably win the Presidency if he ran in 1932 or 1936. He would probably isolate the CSA (further?) and keep it out of WWII. Might of been neutral concerning civil war in Spain.
1939-1946: Long would be unable to carry out his programs (which would already be quite an accomplishment IRL if ever successful) and continues to stay out of war. Assuming USA is relatively untouched in this version of history (then again, no Wilson), the post-war boom takes shape while CSA remains relatively backwards.
1946-1959: CSA would probably request aid and such for modernization from the British in order to keep up with the USA. Assuming a new, moderate President takes hold, we'd probably see mainly that. Now, IRL the Cuban Revolution was evidently anti-Batista. But with no Batista, and the CSA being a bit backwards economically, the Revolution probably would of been stronger. CSA would oppose and probably send troops. Assuming Revolution is a success, morale in CSA experiences a sharp decline and CSA-Soviet relations are impacted greatly.
That's about all I can think of right now.