The US civil war was an internal matter. Americans killing Americans due to a different idea of, bluntly putting it, level of centralization of the union. As brutal as a war can be, there was no other atrocities apart from the usual ''breaches of the Geneva convention*'' we see even today. And in times when the global media was an occasional telegram from over the seas.
Nazism was an international matter. A madman followed by madmen invading other countries in a notion that some races are superior resulting in the destruction of entire generations of Germans, Poles, Russians, Jews...
It brought shame to a country that was also blamed for one bloodbath just 20 years earlier.
Then there was the added notion of neo-nazism springing up in force and letting them have the former symbols to use as a rallying point.
* The Geneva convention was first drafted during the times of the CW, we can't really talk about the actual rules being broken during the CW
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