Hamilton's vision was a Federalist vision. He believed in the establishment of a strong, centralized government that would restore order at home and allow the United States to become a major player in international affairs. To this end, he was a staunch advocate of the Constitution. He was also instrumental in setting up the first federal bank, which allowed the United States to pay off the enormous debts it had accrued during the Revolutionary War.
Hamilton's vision was of the United States as a world power whose wealth would be based on finance, industry and commerce. And it's this vision which closely approximates to how the United States has developed since Hamilton's day.
Jefferson's vision was radically different to Hamilton's. As a firm believer in radical republicanism, he was deeply distrustful of strong, centralized government. To him, this smacked of the kind of tyranny that the American colonists believed had been established by the British. Under the Jeffersonian vision, ultimate political sovereignty resided with the individual states.
Jefferson's economic ideas were also the polar opposite of Hamilton's. He believed that land was the source of the nation's wealth. His ideal was an America of independent farmers, working the land that would provide both economic and political stability. In common with other Republicans at the time, Jefferson distrusted the commercial and financial elites of the East coast, the very people that Hamilton believed provided the key to America's future prosperity.