1. If the North had allowed the South to form their own nation, then slavery, racial, religious, and sexual discrimination would have gone on a lot longer than they did/do in real life. Although many people in the North may not have cared too much if these things continued in the classical South, expansion of slavery to the new States was a real problem, if only due to the political power that would grant the slave owners. And having the South retained as part of the Union led many people in the North to rethink the fairness of fellow citizens in the South being deprived of human rights, given that the victims were in the same country as the Northerners.
2. In practice, the result would have been a relatively poor, agricultural country just south of the more industrial and wealthier North. If you think immigration from Central and South America is currently a problem for the modern USA, I can only imagine to what this kind of division might lead.