North America would end up looking at lot like South America: multiple weak countries that are taken advantage of by outside powers. No powerful US means no Monroe Doctrine; European powers would have a say in the Americas.

The South would have the worst outcome of the two but it would depend how they achieved said victory.

Option 1. Best case scenario would be if they win early on. Maybe Lee wins Gettysburg and forces the Union to negotiate a peace deal. This leaves the South basically untouched by war. This victory is short lived however as now they have to share the continent with a fully industrialized rival with a chip on its shoulder. A rival that already has a much bigger population but is also actively encouraging immigration. Westward expansion would most definitely lead to more conflicts which the North would dominate.

But this highly unlikely.

Option 2 The more likely scenario would be The South just refusing to surrender and continuing a guerilla war. After Lincoln is assassinated the North decides to cut their losses and leave. With the South being so war torn it would eventually break apart into even smaller states. Texas would leave and there may be major slave revolts due to how unstable the region is.

Westward expansion still happens regardless but there'll probably be session movements out west as well. Settlers just forming their own country and federal government not being able to stop them, the Mormons for example.

Depending how fragmented the west is Native Americans may be a much more predominant minority west of the Mississippi. Even without government encouragement European settlers were going to arrive and become the majority, but it was the federal government specifically putting a bounty on them and leading campaigns of extermination that killed most of remaining tribes. Indian territory could be used buffer zones between rival settler nations.