How did Jefferson and Hamilton's ideas influence government policy?

Jefferson and Hamilton each had competing visions of what they wanted the United States to be. Jefferson's ideal was of a republic with weak Federal government (in domestic but not foreign affairs). The new nation would be based upon a largely agrarian economy. Jefferson, as a gentleman farmer himself, not surprisingly came to champion rural economic interests against a rising tide of commercial and industrial expansion.
But his passionate commitment to this ideal wasn't simply self-interested; it was also deeply principled. Jefferson believed that landowners such as himself formed the bedrock of any successful system of republican government. Those who owned land had a greater, deeper connection to the soil; this meant, among other things, that their economic interests were more stable than those of merchants and businessmen. As landowners had more of a stake in society, they would be the ones relied upon to exercise the limited powers of government wisely, instead of sacrificing the common good for their own narrow self-interest.
The best way of preserving this republican ideal was by strongly supporting states' rights. The republican spirit was born out of a war against what the Founding Fathers regarded as the tyranny of the British. The last thing Jefferson wanted to see was the establishment of a remote, centralized government of the United States, essentially re-constituting the spirit of British rule on American soil.
Hamilton's vision was almost diametrically opposite to Jefferson's. He saw America's future as a commercial, trading nation, opening itself up to the world's markets. He wanted the establishment of what we would today call a capitalist economy. Hamilton, unlike Jefferson, was a firm advocate of merchants' and financiers' interests. This strongly influenced his views on the function of central government.
In order for the United States to become the economic powerhouse he envisaged, it was necessary, thought Hamilton, for the power of the Federal government to become more powerful, more centralized. Hamilton advocated the establishment of a Federal Bank that would provide much-needed credit to the growing commercial and mercantile sectors of the economy as well as redeeming debt. This would ensure, it was believed, that the United States would become a trusted trade and business partner in the international marketplace. Centralization of government was a necessary condition of America's future prosperity.
These competing conceptions clashed in Jefferson's and Hamilton's radically different ideas of constitutional interpretation. As a strict constructionist, Jefferson vehemently opposed the establishment of a Federal Bank, not just because it could damage rural interests, but also because it wasn't expressly authorized by the Constitution.
Hamilton's approach was more flexible. By contrast to Jefferson, he saw the Constitution as giving implied authority to Congress to put into action powers that were indeed explicitly granted by the Constitution, such as the power of government to raise taxes and borrow money.
The differences between these two giants of American history have lived on ever since. We can see, for example, how the respective positions of the Union and Confederacy of the Civil War were greatly influenced by Hamilton and Jefferson. Hamilton's vision lives on today in the shape of the world's largest economy and sole remaining superpower, in which the power of Federal government has expanded beyond anything anyone in Hamilton's era could possibly have envisaged.
However, the spirit of Jefferson and his radical republicanism is far from dead. So long as Federal government continues its seemingly never-ending expansion there will be those who will evoke Jeffersonian principles in recalling Americans back to the principles on which their republic was built.
http://www.let.rug.nl/usa/outlines/history-2005/the-formation-of-a-national-government/hamilton-vs-jefferson.php