how do the differences between jefferson and hamilton lead to different views about the role of government

Jefferson and Hamilton diverged in what they saw as the appropriate size and role of the Federal Government in America's new Federalist system.

Hamilton believed that a strong Executive and Federal Government was needed if the US was to be successful moving forward.

Jefferson, ever wary, opposed a strong Federal executive even though he would later as President unilaterally expand the United States with the Louisiana Purchase. So, in many ways Hamilton would win but Jefferson's view would haunt American government for years to come.