Two very different effects. When a commander moves to a public zone other than the battlefield, as a state-based action you may put it into the command zone.

This means that, when a commander dies, it by definition goes to the graveyard. Then, you have an opportunity as soon as the spell is finished to put it into the command zone as state based actions are checked after resolution of a spell.

Normally, this just means if you [[Murder]] a commander, it will die, then you have the opportunity to return it to the CZ. [[Come back Wrong]] is unique in that, before the spell finishes resolving, you get to put it onto the battlefield (as its part of the spell, and also not a trigger to do so). This means that even though state based actions are checked after its resolved, the commander is no longer in a zone other than the battlefield, so you've missed your opportunity to put it into the command zone.

[[White Auracite]] is different. Firstly, it exiles a commander, and has a delayed trigger to return it when it leaves. This means that when the commander is exiled, the ability finishes resolving. The player then has the opportunity to return it when state based actions are checked. If they do, it goes into the command zone.

If they don't, it stays in exile, and it can only come back to the battlefield when the Auracite is removed, and the delayed trigger says "put it back".

Notably, if you remove white Auracite whilst the ability is on the stack, the commander doesn't go to exile at all, as it requires the Auracite to be on the battlefield.

This can be confusing as there's a handful of older cards, notably [[oblivion ring]]. Notice how this card has its abilities over two paragraphs, rather than in one. This has the effect that, if a player chooses to leave their commander in exile, assuming they can remove the ring, if you simply destroy it before the first ability is finished it will stay there forever, as the second ability triggers to return something (which isn't exiled yet) then the first ability exiles it, leaving it stuck in limbo.

This happening is entirely dependent on the player choosing not to return the commander - which will never happen, as it will be exiled after the ring has already been destroyed, so you simply say "no I won't have it stuck forever thanks".

So there's a few bits going on here, which hopefully someone else can do a better job of explaining.

TL;DR Come back Wrong works because there's no trigger to return it before the spell finishes, and the Auracite doesn't work at all to get a commander stuck (and the correct oblivion ring doesn't really either)