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After the war, almost EVERY state - Even the South, was already against having Slavery continued. The rest followed suit within short time. The Federal Govt. also seized a lot more control from the States (prior it was like the EU where the "States" were like the EU Countries, but now it is one "Country" with 50 divisions within). And with that power, the Feds also enforce more equal treatment from States onto other States, as well as States onto other States' Citizens. But the most "significant" change people tend to see is the removal of Slavery - a very important action, but not the only change. Which is why people sometimes claim it wasn't about "Slavery".
I hope this helped somewhat. It is a very tricky topic in general and a lot of people take sides instead of understanding both sides and their motivations.