Distillation of aniline/water mixture
Can't we just distill the aniline off?

The boiling point of aniline is around 180 °C. Remember that water boils at 100 °C.

Do water and aniline mix?

Yes, but the solubility of aniline in water is rather low (around $\pu{30 g/l}$). 

What's the advantage of the steam distillation?

Aniline and water form an azeotrope with a boiling point around 98 °C.