Well, probably no Louisiana Purchase for one. I've read up on every candidate in that election, and I can honestly say Jefferson was the least likely bloke to take Nappy up on his offer.
Whether or not we'd made our alliance with France (which could butterfly away our special relationship with them!
) and gone to war with Britain in 1807 is pretty much up in the air, IMO. Jefferson shared Burr's pro-French (or perhaps just anti-British) sentiments, but he was also a staunch isolationist, so it really depends on whether ideology trumps practicality.
With no War of 1807, we probably wouldn't have conquered Canada, though we probably would have grabbed it later on. But with no war in '07, Britain may not have been so overextended as to have been forced to sue for peace with France (which I'm sure one of the limeys on the board will thunder about for a few posts. I wonder if they'll ever get over that "You bloody Yanks stabbed us in the back!" mentality they always take on when we discuss this era?).
On the plus side, no War of 1807 may mean that the Brits never humiliate us by burning down D.C. Oh well, we won in the end
(though I know a lot of people who still say that us absorbing Lower Canada wasn't so much of a "win"
)
No Burr presidency might also mean that term precedent may have been different. I recall Jefferson writing in his memoirs (which were rather bitter in some places) that had he won in 1800 and gone on to win in 1804, he would have only stayed in for two terms like Washington. As we all know, Burr was POTUS until his death in 1836. One wonders what effects this would have had on Andrew Jackson and his political career. No "Jacksonian Revolution" from '36 to '45, perhaps? Or maybe there'd be one ealier on without a monolith (or, as some might say, dictator) like Burr dominating the political system for so long. Though it's not like Burr's family was immediatly defanged either, so...