Having trouble deciding who to give it to? Which is better?

Blakc Star is WAY more useful, but it depends on how you want to role play it, I quite like azura and so I forfeited the Black star in order to obey Azura

Technically, the Black Star is better.

I take Azura's Star so I can have actual Daedric artifacts.

The Black Star.

The Black Star. Although if there's a mod allowing me to get both variants...

White Star is essentially a reusable Grand Soul Gem
Black Star is essentially a reusable Black Soul Gem, but you can put any type of soul in it.

Depends on the role-play. My assassin who collected Daedric artifacts went with the Black Star. My current character, an Ashlander and Azura worshiper, went with the Azura Star.

You can't capture human souls with azuras star?