Angry Grimace
Two cannibals are eating a clown. One turns to the other and says "does something taste funny to you?"
Seems i fucked up massively..
The Black Star just gives a better gameplay benefit because all Black Souls (i.e. Forsworn or random bandits) are considered Grand Souls. Since Azura's quest doesn't seem to have any level restriction it means you get Grand Soul access faster since at least up to Level 25, I haven't even encountered a Grand Soul bearing creature.
For the Black Star:
when you retrieve the Star, give it to Nelacar in Winterhold. He will fix the star but it can only accept Black Souls (i.e. NPC souls).
For the regular Azura's Star:
take it back to Azura's Shrine. Azura herself will fix the star after the same scene. The Azura Priestess at the shrine will also become a follower. The Azura's Star is still a reusable soul gem, it just works on creatures.
The final gameplay event in this quest
also seems to be one of only a few places so far I've found Daedra Hearts (which you use to create Daedric Armor). It doesn't seem like there are many Daedra in this game beyond this and the Mehrunes Dagon quest. For whatever reason, I chose the regular one because I wanted to hear Azura's dialogue since I always find the Daedric Prince dialogue amusing.