Sounds like some of my old campaigns. My brother and I have each had a few sessions where players were killing each other due to 'unresolveable differences'. Thing is it usually isnt the characters differences, its the campaign itself.
Don't get upset with them behaving how you wouldn't, just find some of the things they find entertaining so they are provided with better options. If the party has more fun proceeding with the quests, earning titles, or getting loot then killing each other may not seem as attractive. Do a needs analysis on the characters and players (they are two seperate entities). A character will seek a reward that satisfies a player's desires in a manner they can achieve while maintaining their character's personality.
You can remove instigators of fights - something my brother has to pick up on still. If the players fight when they get dirty, then it's just asking for it when you make a mudpit they have to move through. Having organizations give players conflicting objectives with quests is another instigator. Tie their quests together so they can both accomplish their goals at the same time. There is no reason you can't use instigators (gremlins stealing items from one player and stashing it in the others backpack is classic) - but these instigators should be used sparingly and only for story related instances. I find it helps to use instigators during a combat, for example: finding potions are missing when they go to reach for them, they argue for a bit, another combat starts due to their yelling (possibly with more weird things going on that clues the players in). The combats give the players a chance to cool down and end the arguing, moving onto something more productive.
In a very sucessful campaign I just ran there was a new rule I experimented with: No one can reroll their character. If you die and the party refuses/can't find some way to bring you back, you'll have to wait for the next campaign. Given a sense of mortality, the players started to care if they pissed off each other. The campaign world was dangerous enough that they accepted they needed each other, even if they didnt like each other. There were a few instances of players wanting to kill each other because of some drastically conflicting personalities, but it never went past some in character threats: "Man I hate how cowardly you fight... You'd be so dead if we had some way to deal with the invis mages"... "Yea, but you don't so suck it up". End of the night everyone had a ton of fun every time!