The question is whether Burr kills Hamilton or not. COntrary to the popular myth which recently sprung up, Burr hated Hamilton as much as Hamilton hated him. Burr was the one who instigated the duel, and he was actually aiming at Hamilton, contrary to the legend of Hamilton shooting a man aiming into the air. So Burr surviving probably involves the death of Hamilton.
The next question is where Burr can go from there. Hamilton went into the duel as a national figure, and he had the potential to go higher, as history shows he did. His hand is evident on the American industrialization and expansionism which marked the 19th century and beyond. Burr was different. His career was past it's pinnacle. He lost a shot at becoming governor of New York, and he failed to become president. Jefferson, and by definition the democratic republicans, will hate him; and the federalists will not take kindly to him killing Hamilton. Still, Hamilton's fortunes seemed to be declining, so a recovery may be possible.
So the United states will face two fundamental changes: no Hamilton, and a living Burr. A surviving Burr brings up the possibilities of an earlier feminist movement; recently discovered documents suggest he was commited to that principle. SO if Burr survives and prospers, might a femenist movement have achieved concrete results earlier? If so, when? On the other hand, if all those conspiracy theorys that get dragged up on here on occasion are true, is it possible for Burr to head southwest, and attempt to carve a kingdom out of Mexican land before the US got there? Perhaps Jefferson and Burr's nations battle it out for the future of North America, and, if this is anything similar to otl, the world? But losing Hamilton will be equally interesting. Who do the Federalists rally around to replace him? Can they achieve the electoral success that they did in OTL? When, if ever, will the US industrialize? Let's assume that Burr is quiet for a few years, and the Democratic-Republicans take control of the country for a few more years. Will they be able to sway themselves from jefferson's isolationism and pacifism when impressment of American sailors becomes an issue? And if they fight, will they achieve the success the USA did in OTL? Many today do not realize how much of a close run thing the war actually was. I doubt that, when war rolls around over the Chesapeake affair (or later, if Jefferson can postpone it, things in that specific timeframe are always a dicey subject for alternate historians), the US could match the stunning successes it made. it might even be concievable for the British to annex sections of the frontier, or for New England to break away (I beleive there were some rumors of it), instead of the string of American victories in Canada.