This is one of the greatest "what if" questions of all. I agree with most of pohnpei's points on the future of the two countries, but I'm not sure that President Abraham Lincoln would have ever given up the battle to restore the seceding states. Remember, had the Confederates somehow forced a Union surrender, Lincoln would probably have never been assassinated. Of course, he may also have never been reelected for a third term, but I don't see him ever giving up the fight to restore the Union.
I also agree that slavery would have eventually died off on its own in the Confederacy as it did worldwide. The Confederate capital would have almost certainly been moved at some point, since Richmond was far too close to Washington D.C.--a temptation that Lincoln could not have resisted. Or would the U.S. have moved Washington to somewhere safer from Lee's conquering armies?
For an enthralling look at such a proposition, you might want to read sci-fi writer Harry Turtledove's mesmerizing novel of speculative fiction--"The Guns of the South." In this tale, South African white supremacist resistance fighters (of the real-life South African AWB) from the 21st century future go back in time (handily, they have a stolen time machine) with thousands of AK-47 automatic rifles to resupply the Confederate armies. Guess what happens? Well, just imagine the Army of Northern Virginia (and Beford Forrest's cavalry!) armed with rifles that fire 30 rounds in a a few seconds. The South Africans even bring heart medication for Robert E. Lee. Southern elan is revived and... you'll have to read the rest.
In my opinion, the impacts of such an outcome on world history would have been tremendous.
Most people would think of the continuation of slavery as the major impact. I don't think so. I think that slavery would have died relatively soon (perhaps, like Brazil's, in the 1880s) because of economic pressures.
What would have been really important, I think, is that the US would never have become a superpower. This is because the US would not have been this monolithic country, pretty much impervious to invasion.
Imagine, instead, the CSA siding with England in WWI while the USA sided with Germany. It seems likely that the two countries would have taken different sides in that and any other war.
It's hard to imagine what the ultimate outcome would have been. Maybe Britain would have remained as the major superpower without the US to challenge it.
At any rate, I don't see how the US (or the CSA) could have become a superpower in such a situation because each would have been too vulnerable to attack from the other.