By nature, aluminum is very lightweight.
While this might not seem like a big deal to the casual shooter, if you carry a lot of ammo, or are hiking/traveling by foot on a hunt, the weight advantage can be significant.
Like brass, aluminum is less corrosive than steel.
Like steel, aluminum does not typically cost as much as brass.
All in all, aluminum-cased ammo provides a strong, durable, low-corrosive, and cost-effective ammo option.
Compared to brass, aluminum-cased ammo tends to be less expensive.
Both aluminum and steel are less malleable than brass. So, in some guns, they do not create as tight of a seal and run less clean than brass.
Aluminum has a lower melting point than brass, so its structural integrity does not lend itself to reloading. Also, because aluminum is a more brittle metal, it is not flexible enough to be easily resized (which is needed if you want to reload a case).