Why does a rooster crow, but a crow caws?

The two terms are both of imitative origin, crow appears to be older in usage and probably for that reason first applied to a common, domestic animal.

Cockcrow dates back to the 1350-1400 period;( Middle English), from Old English crawe, imitative of bird's cry.

Caw was first recorded in 1580-90; of imitative origin.

(Dictionary.com) (Etymonlne)