My take is that Pate is right and they both were companions once and probably had been treasure hunters of a sort, cooperating long enough to even have a "shared" haul. They then had a falling out because Pate rightly deduced that an area was trapped and refused to go inside, Creighton then went inside and fell for the trap.
While Pate is honest, he isn't a good person and I'd say he's more neutral in his morals. After Creighton got caught in the trap he decided not to risk his own skin to save Creighton and instead took their "shared" haul for himself. This combination of being trapped and then Pate taking all of their treasure caused Creighton to blame Pate for the trap even though in all likelihood Pate didn't trick Creighton and he probably told Creighton that he thought it was a trap just like Pate later tells the player.
This one event causes both men to turn on one another, since both think the other's actions were that of a true psychopath. Pate feels that Creighton is overreacting to a trap he told him was there and Creighton feels like only a heartless psycho would not help him get out of the trap. The bottom line is Pate is not the kind of persons that would risk his own skin to rescue another and he assumes everyone else in the world would act like him in the same circumstance.
By the time the player gets to them fighting both are so determined to win this conflict that they both have really stopped caring about their shared treasure haul and are greatful enough to the player for helping them that they let the player keep the treasure. Pate in all probability boobytrapped the chest, since the key to the area with the chest has Pate's name on it. Pate says what he does because he's directed you twice into areas that were trapped and figured you'd handle it, Creighton says what he says because he knew that Pate probably trapped part of their shared haul and that's part of reason why he was so free with handing over the key. Both were kind of hoping the trap would get rid of you since neither really cares about the wellbeing of the player once they got the player to help them end their dispute.