These differences begin with who they thought should govern and what type of government was the best. Hamilton thought we should have a strong central government in the interests of commerce and industry, while having the national government in charge. However, Jefferson felt that the people should rule with a decentralized, agrian government in the terms of freedom and the people should rule themselves. They also had conflicting ideas for what economy suited us best, Hamilton believing it to be industrial and Jefferson believing the best was agricultural.
Along with those conflicts, they didn't agree with how the constitution was to be interpreted. Hamilton was a loose constructionist, wanting to stick closer to the thought of the central government ruling. Jefferson was a strict constructionist, believing that the constitution was to be followed closely.
Lastly, their difference in ideas helped formed political factions. They became two sides, the Federalists and the Antifederalists - or Republicans. Alexander Hamilton's side was the Federalists, they stood for the urban mercantile interests of the seaports. Thomas Jefferson's was the Republicans who represented the southern and rural interests.
Alexander Hamilton and Thomas Jefferson had very different political views, which is why our first president, George Washington, had them both in his cabinet. Hamilton was the first Secretary of the Treasury, while Jefferson was the first Secretary of State.
These differences begin with who they thought should govern and what type of government was the best. Hamilton thought we should have a strong central government in the interests of commerce and industry, while having the national government in