Both lateral roots and root hairs grow outwards from the root. The difference lies in the origin, i.e. where the structure starts.

Lateral roots are said to originate endogenously because they originate from the pericycle, which is deep inside the root.

(Image source: Lateral Root Emergence)

Root hairs are simply outgrowths of epidermal cells. Since they are so superficial, they are said to originate exogenously.

(Image source: Monocot Root: Epidermal Root Hair in Lilium)