Why does Annie have an Autoattack range of 625?

Probably to help with conserving her stun until needed by providing her a safer auto-attack range for farming.

and why do similar characters like Brand and Veigar not have the same or similar range?

Brand has more than that kind of range on his abilities.

Annie's abilities have pretty short range by mage standards, and her passive requires she actually doesn't use her abilities in many circumstances other mages can and will. Where Brand, Ahri, Lissandra, Veigar, Fiddlesticks, Orianna, etc. all have their CC tied to 1 button and can utalize at the drop of a hat to either capitalize on an oponent making the wrong move or to save themselves, while farming waves with their AoE skills - Annie doesn't have the same kind of reflexive options. She requires more foresight to access her CC, and using her kit's power comes with greater trade-offs.

Pretty sure that's the logic behind it. She's had that range out of Beta so it might be outdated in the current meta. But Riot also placed some draw-backs to her AA range. For one, Annie doesn't have any AD scaling at all. Secondly, her AS and AD scaling actually puts her at the worst auto-attack damage of all 100+ champions in the game at 18, and for most of the levels leading up to that (I think Karthus is the worst at level 1).

If only Lux could have that long range.

Would make her a lot better.

Why do Brand and Veigar have non-ult spells that can hit you from 900 range but annie can only hit you from 625 range? Seems kinda imbalanced if we ignore everything but this.

So pedophiles can't reach to her, yes man in the Bear suit, i'm looking to you.

I don't remember AD Annie was viable.

Because she is a sorceress who commands her attacks over great distances!!!

On a more serious note, I - have - no - idea. Her auto animation blows though.

Her ability range is short, and her CC is not always accessible. Range is justified.