Genuinely what is it about burr that makes people act in insane obsessive ways. Obviously we have hamilton throughout half of his political career but also Jefferson during the treason trial. That man wanted burr dead so damn bad. Did he have an annoying face? Was it his clothes?
the boring (and oversimplified) answer here is that it was because burr didn't really adhere to either political extreme (DR vs federalists) too strictly. he kind of floated between the two. whether you believe it was because he was just trying to be honestly impartial or you think he was a political opportunist is up to you.
to me, it's always seemed like burr really did have ideals and principles, but they didn't perfectly align with hamilton's or jefferson's and tended to be much more focused on social causes (progressivism, broadly) than strictly policy-based (big government & mercantilism vs. small government & farming) -- so he floated toward whatever party he thought would effectuate his personal, progressive views. this, of course, translated into burr being a man on his own, with no ideological party allegiance. he loved politics and politicking / canvassing / reaching out to constituencies, but he was never a policy man. this was almost unheard of in burr's time (drumming up support for your party was seen as ambitious and uncouth).
jefferson's relationship with burr started out as a friendship between two people who had basically the same kinds of progressive views. i do think, at some point in the early 1790's, for a hot second, they had a genuine friendship. i think burr actually really did like jefferson, but i think jefferson saw burr more as an asset to political power in NY. it's really important to stress burr's new yorker status-- a man who was, also importantly, on personally friendly terms with hamilton. jefferson would have known this, and it would absolutely color his view of burr. but because of their (basically) similar politics, they were able to work together fruitfully. of course, once burr became a liability, and stopped being useful, jefferson dropped him and distanced himself from burr. it was especially easy to abandon burr after the duel. (seriously, july 12th, 1804 was like christmas for jefferson & his adherents.) this obviously made burr more desperate for power, culminating in the burr conspiracy. once burr became (in jefferson's eyes) a genuine threat to his administration, that's when things turned ugly.
hamilton's relationship with burr is obviously far more complicated, due in large part to hamilton's insistence that he personally actually liked burr, but did not trust him with political power. and depending on when you think they first met (1772 or 1773 seems logistically the most likely given how tiny elizabethtown was, but for sure by 1776), they'd known each other for far longer than jefferson had known burr -- and probably longer than they'd known most other people in their social circle. to me, hamilton's obsessive issues with burr stemmed more from control and personal betrayal, not necessarily policy or administrative disagreements (although they definitely had those, lmfao). like always, when talking about hamilton's issues with burr it's super difficult to not veer into psychoanalytical territory because hamilton absolutely fluctuates between fascination, projection and revulsion with burr, whereas jefferson just kind of hated the guy and wanted him gone because it would be politically expedient. it was hamilton who started the rumors about burr's "deviant" sexuality and power. which others simply ran with after hamilton continually stressed the fact that he knew burr better than anyone else. (as an aside, if we're to understand that hamilton himself was not straight, and was sexually attracted to men, this makes his professed "expertise" about burr, and his explicit campaign against him, all the more noteworthy. but that's a different post)
to put it simply, jefferson's obsession with burr seemed to be more circumstantial than pathological. hamilton's was the opposite: it really didn't matter what burr did, he was always going to get under hamilton's skin for reasons that none of us will ever fully understand. anyway, as always, the truth of it is probably somewhere in the middle.