Yes, the personalities of Yin and Yang Kurama are different as it is based on the principles of yin and yang. To balance everything, Kurama has both good and bad sides that are split into two parts. Minato, while sealing him inside Naruto, made sure that the “good” side of Kurama goes to Naruto.
The reason he did was because he didn't want the "bad" Kurama inside Naruto, as we know for a fact that Kurama is the most impatient and angry biju of all. Another reason is because Kurama had loads of chakra. Naruto as a child, even being an Uzumaki, cannot hold that much inside him (that's why Minato split it).
Naruto struggled to gain the trust and friendship of "good" Kurama. It would gave been a lot more tougher if it was the bad and angry "bad" Kurama will be. It is also seen that the "bad" Kurama was surprised that Naruto was able to befriend the "good" Kurama. You can hear him say:
"Looks like Naruto sweet talked my other side."
Since Naruto has already befriended the "good" Kurama, the "bad" Kurama didn't hesitant to give chakra when Minato shared it with Naruto.
Despite Yin Kurama being bad, he does know how to compliment someone and addresses the wielder by his name. Yang Kurama doesn't anything about compliments and he initially addressed Naruto as "brat":
- "That was quite the touching speech your son just gave there... Minato"
- "He became a nice brat that really loves his parents... Minato"
Moreover, Yang Kurama was surprised that he was able persuade his other half (it might mean that is difficult to reach his thoughts and feelings):
- "He was even able to persuade my other half"
- "(that was) Enough to move even this(other) half of me"
Both of the Kuramas shared their chakra on the basis of the mental state of the wielder. Yang Kurama's chakra automatically mixed into Naruto's when Naruto was full of hate, anger, and dark thoughts. Minato had access to Yin Kurama's power while being pure hearted, happy, normal good guy.