What are the advantages and disadvantages of each version of Azura's Star?

As far as I understand it, the only thing that matters for enchanting is the level of creature or human you soul trap. So it doesn't matter if you use a human soul or a creature soul.

However, nearly all soul gems only trap creature souls: only black soul gems will trap human souls. And black soul gems are pretty rare. Black soul gems (including the Black Star) work on humans of any level, including common NPCs like bandits. So if you're looking to expand your soul trapping opportunities, having Nelacar repair it is prudent.

However, siding with Nelacar will prevent Aranea from ever becoming a follower, which might not be what you want. While it's stated to only hold Black Souls (Humanoids) it can actually hold all types of souls, making it a straight up upgrade, as long as you're willing to go that way in the quest line.

In terms of a quest to collect all the Daedric artifacts, I'm not aware of any in game. However, if you're an achievement collector, all three platforms have:

Oblivion Walker

Collect 15 Daedric Artifacts

But both versions of the Azura's Star count for it, so choose whatever you want.

And as an aside, there is an Oghma Infinium quest, just not in the same style as Oblivion:

The Black Star combined with soul trap weapons can be great for leveling up your Enchanting Skill. Well, as long as you don't have the unofficial patch installed which "fixes" it.

Every time you kill something or someone with your soul trap weapon and it fills the Star, turn around and use the star to refill the weapon.

Azura's Star is of less use in this endeavor since it can't capture black souls(.75x the XP of enchanting an item with both the Mage Stone and a simple rested bonus active).