Not only have there been a lot of threads on this board, discussion of the (misnamed) American Civil War on this board have also pretty comprehensively established that once the Confederacy starts a war with the rest of the United States, they aren't going to win.
They really seem to have been insane, or thought the northern states would fight a couple of battles and then give up. It doesn't make sense. Though its worth noting that the Confederacy that fired on Fort Sumter were just seven states, and there were fifteen slave states, and a disproportionate amount of the leaders of the actual war effort came from the states that had not signed up before Fort Sumter and who were not involved in the decision to start the war. The war effort was conducted fairly sanely and competently, the decision to fire on Fort Sumter, not so much.
If the rest of the United States decides that the Confederacy is in rebellion, and decides to fight and stick with it until the rebellion is crushed, which is what happened, then the Confederates are fighting the world's leading industrial power with just two factories in 1861, and their opponents can just march armies south and concentrate entirely on their opponents.
The only "Confederate independence" scenario that makes sense is one without any war at all. The seven original states just secede. The POD is that the federal government this times agrees to negotiate, evacuates their posts within the Confederacy, and eventually lets the seven states go; or the Confederates leave the remaining forts alone, and with the Constitution silent on secession the northern states have no justification for war themselves and eventually come around to a negotiated separation.
In this scenario, VIrginia, North Carolina, Tennessee, and Arkansas do not join the Confederacy, Virginia and Tennessee having considered and rejected the idea before Lincoln's call for volunteers.
Slavery will remain legal in the United States for an embarrassing long time. The only slave state that had a serious abolitionist movement in 1860 was Missouri. There may even be a scenario where the border states industrialize and rely more on paid labor, but though there are not many slaves, slavery remains on the books as legal. IOTL, states keep a ton of ridiculous and outdated laws on their books pretty much forever. There are not only no 13th, 14th, and 15th amendments, there might be a 13th amendment guaranteeing the right of a state to keep slavery legal (such an amendment IOTL passed Congress in 1861 and was ratified by at least one state).
The CSA will survive. It will cover a large territory and will have British support. It will basically be a backwards British protectorate. Eventually Russia will champion some sort of international abolitionist movement, and an embarrassed British government will pressure the CSA and Brazil toward abolition. Slavery could even be abolished in the CSA earlier than in USA. After the adoption of the 13th amendment, the constitutions of both countries will state that any abolition has to happen at the state level. Abolition will be compensated, which pretty much happened every time it was done peacefully, which was everywhere but the USA and Haiti.
Free African-Americans in both the USA and CSA (New Orleans had a sizeable population of them before 1861) will be better off IOTL, the efforts to keep the free Black population at the lowest status possible was really an after effect of the war, as shown by experience in other countries where abolition was done peacefully. There will be alot of people born into slavery after 1861 who will be obviously worse off.
Politically, the states remain much more powerful than IOTL, for obvious reasons. The US Supreme Court may remain at six or seven justices. I'm not sure if the CSA ever establishes a Supreme Court. The Whigs re-emerge as the second party in the CSA, and in the USA remain around, probably as the "Unionists" as a major third party. They would have a base in the border states, and with no disputed 1876 election ITTL, there is no formal or informal basis for the two party duopoly.
The Grant administration obviously doesn't happen, but I don't think the other late nineteenth century presidencies necessarily change. Butterflies start to accumulate and affect the twenty-first century. Unlike in many "South wins the American Civil War" timelines, Woodrow Wilson is not a Confederate president. He was born in Virginia and elected governor of New Jersey. Both states are in the USA ITTL.
The CSA will want to expand south, and they have a good chance of getting Cuba from Spain, either peacefully or through ITTL's exclusion of the Spanish-American War. They are a banana republic, but its not as if Spain is in much better shape, so they can pull it off. Its World War I where the butterflies really come in. The CSA probably is pulled by the British into the war fairly early. But I think the USA remains neutral. This version of the USA is much more anglophobic than the one we got.
