So you can get Azura's star or the Black Star in that Daedric quest so I have questions.
Can you absorb black souls with Azura's Star?
Can you absorb regular souls with The Black Star?
Which is more powerful?

Many thanks

Black is the best as it is always grand for humanoids. However, you miss out on a good follower.

You can only absorb white souls with the regular Azura's Star, and you can only absorb black souls with the Black Star. ALL black souls are grand souls, so they're considered better, but siding with Azura also gets you one of the better followers in the game, and is considered the 'good' guy option.

but can you absorb regular souls (e.g. animal/dragon/troll etc) with it as well? I really want something super for enchanting and recharging weapons

Not dragons but it's easy to collect bandit souls and they are always grand.

The Black Star is the best imo, but also consider that once you get higher level you can simply go to the mage vendors and buy all the filled grand soul gems you could ever need for making enchanted items.

So you would really only need it for possibly only topping up enchanted weapons/staffs.
I have had it contain non-grand souls, so that must have been non-human that filled it.

I'd go with which every fits your character roleplay wise.