The South did not intend to overthrow the government in Washington, only to disassociate themselves from it. Accordingly, theirs was not a war of revolution, but one of independence, as per the Rhodesian Unilateral Declaration of Independence of 1965 from the British Empire, or that of India and Israel from their previous colonial master.
Odd that the supposedly free-spirited Australians never went that route....
And the difference is?
Leaving the Union was overthrowing the Constitution, the very reason the gov't had been founded.
The Southern states were members of the Union, they were not part of a colonial system.