Among other things, the Nullification Crisis showed how tenuous the Union was. Angered by the failure of President Jackson to deal with the issue of tariffs and their damaging effect on the Southern economy, a number of important politicians, most notably Jackson's Vice-President, John C. Calhoun, openly defied the authority of the Federal government.
Many had fondly thought that the precise relationship between the Federal government and the states had been fixed by the passing of the Bill of Rights. But the Nullification Crisis showed that that wasn't the case. The states, especially those in the South, still had a distinct identity and were thus more than willing to stand up to the Federal government in defense of what they saw as their inalienable rights.
The aggressive response of the Jackson Administration provided a foretaste of how the Federal government would react thirty years later to the secession of the Southern states in the wake of Abraham Lincoln's election. Jackson wanted a compromise with South Carolina but was prepared to use armed force if necessary to enforce the will of the Federal government. Prescient observers noted that, if a similar crisis broke out in future, then a similar threat of force would need to be made in order to keep the states in line. And so it proved.