A part of our answer to this question must rely on literary and anecdotal evidence. A striking feature of Washington Irving's famous story "Rip van Winkle" is that when Rip awakens after his sleep of twenty years, his feeling is that he has entered a transformed realm—and simultaneously, the people around him see Rip as a relic, a symbol of a vanished world.
It is open to debate how much had physically changed in America between 1776 and 1812. Though the industrial revolution was in full swing, including in America, by the early 1800's, major technological advancement had not yet occurred that radically changed people's daily lives. The steamboat had recently been invented, but gas lighting, the railroad, the telegraph, and photography were all still in the future.
What Rip van Winkle's story attests to is the psychological change that had taken place in the newly independent country. Unlike in 1776, by 1812 Americans no longer thought of themselves primarily as English. They were largely aware that now they were a new people. Yet interestingly, it was partly because of the changes taking place in the Old World (in Europe) during this period that the nature of American society and its manifestations in the world of politics and government changed as they did.
In the period from 1789 to 1815, Europe went through a tremendous upheaval with the French Revolution, its effects upon other countries, and the Napoleonic Wars. The nascent American political world of the 1790s (and thus life overall in the US) was deeply affected by the antagonism in the European world between liberalism—the ideals of the French Revolution, which had been shaped partly by the French participation in the American Revolution—and conservatism in its reaction against not only what was occurring in France, but against the entire Zeitgeist of the eighteenth-century Enlightenment.
This split in thought was as wide in America as in Europe. The new political parties of the US, the Federalists and the Democratic Republicans, were at each other's throats to an extent that makes the "culture wars" of the 1960s up through our own time look almost tame by comparison. In the 1790s Americans became an ideologically oriented people. Questions relating not only to social change but to the nature of the economy—the centralized banking system favored by Alexander Hamilton and the Federalists versus the greater self-reliance of local economies favored by Thomas Jefferson, with his ideal of yeoman farming as the basis of America—became critical in the development of the US.
The country became increasingly polarized between the mercantile and newly industrialized business interests of the North and the agrarian, still largely feudal Southern society. In the North, slavery (a relatively minor part of the Northern economy to begin with) had been, or was being, eliminated through gradual abolition laws. In the South, slavery continued, and the Southern elite became increasingly defensive about their "peculiar institution." By 1812, it had become clear that the unity of the new country was still a fragile thing, and that an internal conflict leading to war was a strong possibility.
The paradox in this ideological split developing in early nineteenth-century America was that the Federalists, the party initially siding to a large degree with the conservative elements in Europe in the 1790's and wishing to regard the War of Independence as simply having been a "family quarrel" with Britain, formed the core of the progressive elements in America that wished to abolish slavery. Though Jefferson endorsed the most radical liberalism in the changes represented by the French Revolution, he and his party (in spite of Jefferson's repeated statements throughout his life that slavery was wrong) were the ones who perpetuated the system of slavery, the basis of the South's primarily agrarian economy.
In 1803 Jefferson, through the Louisiana Purchase, added a huge stretch of territory to the United States. Unlike when independence was declared, the US was now not primarily a chain of coastal states but a country with a continental destiny and an increasing awareness of unlimited vistas and of the potential to become one of the major powers in the world. This new awareness represented a basic change in the psychology of a people whose ancestors had settled the New World chiefly in order to escape the perpetual conflicts and injustices of Europe.