Why is the civil war the clearest example of a conflict between state and national governments?

Because there was dispute on whether the United States was a single entity or an amalgamation of independent nations. Before the Civil War, some states, like South Carolina, thought that a state had the right to nullify federal laws and even to secede from the United States.

When the original 13 independent colonies announced their independence from Great Britain in 1776 they considered themselves as independent states, however, the requests of the Revolutionary War obligated the states to acknowledge a need for a central government.