Thomas Jefferson and Alexander Hamilton differed in that Hamilton believed in the idea of a strong government with huge oversight with voters limited to those of appropriate educational backgrounds, while Jefferson believed in the idea of a small government with minimal oversight and with common men as voters. The two politicians were responsible for the creation of the Federalist (Alexander Hamilton) and the Anti-Federalist (Thomas Jefferson) parties.
Perhaps the largest difference between the two politicians was their view on voters. Hamilton believed that voters should have a strong educational background, be land owners and be wealthy. Jefferson believed that the common man should be running the country and felt that every person was an equal, regardless of his educational background, land mass and wealth. Hamilton was also a believer in "big government" and thought that the people needed to be controlled and ruled by a large government that knew what was best for them. Jefferson felt that the people could role themselves and the government should favor individual rights.
Jefferson was also against many federal roles, as he wanted government to be as small as possible. Jefferson, therefore, did not want a federal bank reserve system, while Hamilton did. The two politicians helped to solidify the United States political system in their many debates with one another. In fact, the dual party system is a result of the two factions that these politicians created when they were advisors to the first United States president, George Washington.