Spanish native speaker here. 'CASTAÑO' is a term used to describe BROWN hair and eye colours, regardless of how light or dark that shade of brown may be. 'MORENO', on the other hand, is used to describe someone with BLACK hair or a dark skin colour.

castaño is an expression used only for hair color or eye colour and is refered to the colour of the chestnut. moreno can be used to skin colour, eye, hair etc we can say that pelo moreno is basically the same as pelo castaño.

También educadamente cuando se refiere a una persona de raza negra se le dice "moreno" o también se refiere a una persona de piel canela o bronceada.

in general you use "castaño" for the hair, "moreno" for the skin when you describe a person, and cafe/marron for the objects and things. On Argentina, for example, all people with black hair is "moreno/a".