A more cautious personality might mean that they are more hesitant to engage in battles or take risks. This could lead to different outcomes in some of the key confrontations in the RWBY story. For example, when facing Salem, instead of Ozpin's more aggressive (in some ways) approach of trying to keep her at bay, the reincarnated character might focus more on defense and building up their own forces first. They might also be more secretive about certain things if they are overly cautious, which could cause some misunderstandings among the other characters.
If the reincarnated character has a more outgoing personality compared to Ozpin's original reserved self, they might interact with the students in a very different way. They could be more hands - on in training, more like a friend and mentor rather than a distant figure. This would change the relationships between the characters in Beacon Academy for sure.
On the other hand, if the reincarnated character has a very impulsive personality, they could rush into things without fully thinking them through. This could lead to some disastrous consequences. For instance, they might prematurely reveal important secrets about the world of RWBY or make hasty decisions regarding the use of the relics. However, it could also lead to some unexpected and exciting plot twists as they try to deal with the fallout of their impulsive actions.