Slavery issue aside, had the Confederate states been successful in establishing their independence, it would have opened the door to future secessions. We would almost certainly have had a separate nation on the West Coast, for example (California and Oregon were talking about just such a thing). Before long, the United States would have totally disintegrated and we would have ended up looking like Central America - a bunch of failed republics ruled by despots and military leaders.
It's a damn good thing we were able to reunite this country and put the Civil War behind us (at least most of us). Otherwise, we wouldn't have been in position to stop Hitler and Imperial Japan in WW2. The world would be a totally different place today - for the worse.
President Lincoln and the loyal people throughout the country saw the folly of separating.
President Lincoln once even referred to the example of Europe as something to be avoided.
What these "self determination" fanatics fail to see is exactly what you point out. We are stronger together than we could ever be apart. This is so obvious it makes me think that what the neo-rebs -really- want, is what really was lost in the ACW -- white supremacy. President had a part in ending white supremacy. That is what galls them about Lincoln, and that is why he is a tyrant to them.
It would be nice to have the freedom of a 19th century "mountain man" like Jeremiah Johnson, but it is just not doable.
Walt
You assume the world would have changed in no other way had the Confederacy kept it's independence - but that is absurd. For example, there might have been no American involvement in WWI, with the very possible result of a more equitable eventual European peace. That would have made the rise to power of Hitler much less likely. The American war with the Japanese was if anything even more uninevitable. It only came about because of FDR's disastrous handling of the American economy following the great depression and his choice to involve the country in war in an attempt to undo the harm he had done.