OK, a few things to think about:
1) How much turf does the CSA win?
2) When do they win the war/how much damage do they take during the war?
3) Is their victory diplomatic or military?
I do not think a CSA with only the core 11 states past 1863 is a viable entity as a major power in any sense past the 19th century and would likely be an eventual target for reconquest by the USA, or exist as a British satellite as a means to harass the USA. Eventually I see it either becoming a sort of Argentina with focus almost solely on agriculture and a chance to gain some industrial capacity but without a great deal of mineral wealth behind it until the discovery of oil. It might lead to an interesting scenario though if, in the later 20th century, it becomes an OPEC member and tries to retake KY, OK, and maybe WV by means of an oil embargo.
Assuming a military victory before or at Gettysburg without some sort of radical change, I figure the CSA takes their core 11 states with kentucky and modern Oklahoma. They begin a crash industrialization program focusing on northern Alabama, northern GA, and perhaps eastern TN later on. Rail transportation and steel manufacturing become immediate priorities and South Carolina becomes a world leader in textile manufacturing within 20 years. There is no Thanksgiving there, slavery persists into the first decade of the 20th or last decade of the 19th century under threat from British sanctions. A beaten US will ally with Germany and, if the UK had anything to do with the CSA victory, eye Canada for possible conquest. Washington will no longer be the capital and it will likely be moved to Chicago or Milwaukee or Philadelphia. CSA adventurers will try to take control of Cuba soon after the war ends and any territory in the Caribbean or Central America will also be eyed for possible conquest/intervention. They might even try to build a Panama Canal and cause the deaths of hundreds of thousands of laborers/slaves, perhaps this is how slavery meets its end?
In the long run I see the CSA aligning with London and the USA with Germany with the likely conflict predicted by Turtledove for World War I. Confederate interests will turn to Pacific access at some point, and a land connection via Mexico would not be out of the question. However, I am not sure that the war will turn as Turtledove predicted and do not know if a conflict in the 1880s would be inevitable. The CSA will build a strong Navy and the USA will likely have one that is at least its equal, but where the CSA goes from there depends on how the rest of the orld unfolds. I do not think the Confederates would be easily reacclimated to USA rule after 2 generations of independence nor do I believe they would necessarily parallel the Germans for a world war II emulation. Should the CSA win a WWI scenario I think they would claim WV (if they don't take it initially), NM, AZ, and at least two of the three MD, MO, or KS. If the UA wins I tihnk they'd take KY, OK, and maybe western TX or parts of northern VA.