Do you bring it back to her priestess or back to the elf?

And does his/her tinkering with it really make a difference?

I never bring it back to Azura, my favorite type of character is a Necromancer, so I prefer having the .

Even when I don't play a Necromancer/Mage, I never give it back to Azura. [censored] Azura, she's the most evil out of all the Daedra.

Yes.

But I will forego spoilers and just say this; the choice you make doesn't have a long term effect, the Star just works slightly different depending on who you give it to. For RP reasons, go with Azura. Not wise to [censored] with a Daedric Prince. Unless your character is an idiot, of course, then deny Lady Azura.

To the guy above me, how the hell is she the most 'evil' of all the daedra? I don't think any of them are truly 'evil', but surely Molag Bal would be more fit for that title, being the King of [censored] and all.

Brought it back to the priestess. Didn't know what it done at the time. Small hidden things like that are great, imo. Reminds me of the days of the hardship of obtaining Daedric armour in Morrowind.

You now have two choices. If you return to the shrine, you will speak to Azura. She will tell you that it is unusable until you have purged Varen's soul from the star. Likewise, if you return to Nelacar, he will tell you the same thing.

The difference is the Star's use afterwards. If you bring it to Azura, you will be able to use it to trap any size soul except for "black" souls, or souls from other sentient beings. If you bring it to Nelacar, the game states you will only be able to use it to trap "black" souls, and nothing else. However, the Black Star acts just like a normal black soul gem and is also capable of catching "white" souls. Black Soul Gems are the rarest and most expensive of the soul gems, since all black souls are grand souls, so it's more cost effective to take the black soul-trapping version from Nelacar.
Once you have made your choice, you will be transported inside the star to battle Malyn Varen. He has a powerful staff and is capable of summoning dremora. Once you have defeated him, you will be transported back to Skyrim, and the star will be yours. If you chose to have Azura fix the star, speak to Aranea Ienith, and she will tell you that Azura has stopped giving her visions upon your completion of the quest. Aranea will then give you the option of having her as a follower, saying that it will give her a feeling of purpose in life now that she has lost her connection to Azura.

Source -
http://www.uesp.net/wiki/Skyrim:The_Black_Star

i'm playing a warrior that goes more into the good guy side, so i gave the star to my quest giver.

i can tell that there is a difference between giving the star to the priestess or the elf. not gonna go into detail though.