How did Hamilton's and Jefferson's views of government and the economy differ?

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Hamilton, who was George Washington's Secretary of the Treasury, argued for a powerful, centralized state. He wanted a standing army, a national bank, assumption of state debts by the federal government, and high protective tariffs. These measures and institutions reflect the fact that Hamilton envisioned an economy based on manufacturing, one which would allow the United States to rival Great Britain, the nation which Hamilton's economic system fairly explicitly emulated. Jefferson, on the other hand, thought these measures were anathema to a republic of free people. He envisioned an economy and a society based on landholding farmers, whose produce would allow the United States to achieve and maintain economic independence. Rather than producing manufactured goods, which he thought would produce a large, propertyless working class, Jefferson proposed that the new nation should simply trade for many of these goods. Accordingly, he favored a fairly weak central government, one which kept most of the powers of governing at the state level. Perennially in debt himself, he had a visceral distaste for such financial institutions as the proposed Bank of the United States, which he thought unconstitutional in the first place. 

The disagreements between these two men and their followers, which also involved foreign policy, contributed to the development of the Federalists and Republicans, the nation's first two political parties. Many of these issues, particularly the proper scope and extent of federal power, are still current today.