When it came to Thomas Jefferson and Alexander Hamilton, things started out amicably enough, but it went downhill from there. They were different in every possible way, from their beginnings (Jefferson the son of an aristocratic planter, Hamilton the illegitimate immigrant) to their ends (Jefferson died at 83 of natural causes, Hamilton at 49 of a gunshot wound), their personalities (Jefferson was laconic and avoided conflict, Hamilton was volatile and thrived on it), but most importantly their beliefs.
Jefferson believed in a weak central government, and was very suspicious of those in high government positions. He was an optimist by nature and believed that people could govern themselves. Hamilton's beliefs did not "mesh" with his. Hamilton believed in a strong central government and was suspicious of the masses, believing the only way to govern them was by force. Hamilton, in contrast, saw things through a dark filter and was much more pessimistic. They thought the other an intriguer, and thought that if the other's beliefs controlled the government the government would collapse.
In addition, Hamilton's department, the Treasury, was becoming the most influential department and sported a large amount of clerks and others working there. Jefferson's department, State, had few if any clerks by contrast. Jefferson thought that Hamilton, who wasn't even from the colonies, was trying to control the government. It was a thought that horrified him. Hamilton thought that Jefferson controlling the government would be a nightmare with no order whatsoever. This led to more animosity.
Lastly they had different visions of what they wanted the country to become. Hamilton wanted the country to focus more on manufacturing and have a diversified economy, Jefferson wanted it to stay in the country and continue with the quaint country lifestyle.
In short, they hated each other because their beliefs were so different and there was so much mutual hatred between them because of it. Both were brilliant, both would influence the country, and both would be remembered in history (in varying degrees, that is), but they could not and would not work together.
Whoa, that is one long answer. :) I didn't even realize it.