There were some changes between the Confederate Constitution and the United States Constitution, but the most significant change in the relationship between the Federal Government and the State Governments - the rights of States to nullify laws and to secede from the Union - were not incorporated into the Constitution. The reason for this, of course, is that the seceded States asserted that those rights already existed under the United States Constitution, and modifying the new Constitution to explicitly include those rights could be seen as an admission that they had never had those rights when they were part of the United States.

So, in summary, the most significant legal difference between the North and the South was that in the South a state could nullify any law it did not agree with, and the Federal government would accept that nullification, while in the North, that nullification would not be accepted by the Federal government. And a Southern State could secede from the Confederacy without having to fight a war.