Political Differences
The most important difference between the two men were their stances on politics. It is important to mention here that Hamilton was a Federalist (in support of big business, banks, large government) whereas Jefferson was a Democratic-Republican (small government, state's rights, friend of the yeoman). While Hamilton's support resided in New England (New York, Massachusetts, etc), Jefferson found his support in the more southern plantation states (Virginia, Maryland, etc). These political differences came into play due to their...
Foreign Policy and Office
During the duration of the Washington administration, Alexander Hamilton was the second most-powerful man in the government. As Treasury Secretary, he was charged with, and completed, the rearrangement of the American financial system, economy, trade network, tax network, and debt plans (America was ~15 million in debt after the Revolution). In comparison, Jefferson, as Secretary of State, was responsible for leading the newly created United States on an international path to success, ensuring that they had trade deals, dealt with British creditors to settle debts, as well as ensured their allies during the Revolution would stand by them (primarily France). If you see any overlap in their jobs, that is because there was a lot of it. This became an issue about which their animosity exploded on many occasions. There were incidents where Jefferson would deliberately sabotage attempted trade negotiations with the British Crown as he was a bit of a Francophile. Hamilton wasn't particularly cooperative either as he was vehemently against diplomacy with France while the French Revolution was occurring. While the Democratic-Republicans were more Jacobian in their view, the Federalists, being slightly more elitist, were disgusted by the violence that was occurring in France and wanted no part in a populace that would treat the established government in such a horrendous manner. This was yet another thing that differentiated Hamilton and Jefferson.
Domestic Policy
Perhaps the most crucial of differences were their views on domestic policy. While Hamilton believed that the debt of the Revolutionary War should be absorbed by the Federal Government and then paid off by the states, Jefferson's coalition thought that it was unfair; some of the states had already paid off their war debts and therefore shouldn't be made to absorb this new national debt. In addition, there was the entire issue of whiskey, as Hamilton attempted to tax whiskey profits in a move which resulted in the Whiskey Rebellion (Jefferson was, of course, opposed to this).
Background
Another difference, although less important, was their backgrounds. Remember, Jefferson grew up in privileged plantation society. Hamilton came from the slums of St. Croix in the Caribbean. Hamilton went to war as a Colonel during the Revolution and fought, literally, by Washington's side. Jefferson went to France as a diplomat, spending his way through his inheritance even when he was horrifically in debt. When the Revolution was won, they took very different paths. However, they were similar people in many ways.
Similarities
Jefferson and Hamilton were both highly educated. Jefferson studied at the College of William and Mary, while Hamilton studied at King's College (now Columbia University). They were also both intellectuals and men of the enlightenment, feeding off of the ideas of enlightenment philosophers and becoming enlightenment thinkers in their own right. They both founded schools (Hamilton-Oneida Academy, now Hamilton College; University of Virginia). They were also both abolitionists, although Jefferson owned slaves throughout his life. Finally, they were both extremely humble in their own regard. Jefferson didn't have it written on his tombstone that he was president as he deemed it an accomplishment not worth noting. Hamilton would often flee from the public eye, and although he was sometimes arrogant in his writing and speech, he was not one to brag about his accomplishments.
Conclusion
Hamilton and Jefferson were both very different people. They had some similarities, they had more differences, but they were both patriots. They're also some of the most fascinating people in the history of any country, and their rivalry more entertaining and enthralling, and perhaps even more ugly, than the rivalries in our current election (though neither of them had the indecency to go after the others' family).