The major difference of opinion during the Washington administration was between Hamilton and Jefferson. These two men disagreed on many aspects of foreign and domestic policy.
In general, Hamilton wanted a strong national government that would promote an economy that was based in large part on manufacturing. By contrast, Jefferson wanted a weaker national government and an economy based on small farmers. This difference of opinion could be seen, for example, in the battle over the Bank of the United States. Hamilton thought the federal government had the right to create such a bank (an expansive reading of the Constitution) and that the bank would help to create a more stable economy. Jefferson thought the federal government had no right to create a bank (a restrictive reading of the Constitution) and that the bank was not necessary for the economy.
This sort of argument over the power of the national government and the proper nature of the economy typified the differences of opinion within the Washington administration.