The UK has a similar distinction. "Hackney carriages" (the classic "black cab") are allowed to pick-up passengers on the street, while private hire vehicles (often called "minicabs") can only be used by pre-booking. The former are more regulated than the latter. Ubers are minicabs.

(Black cabs don't have to be black, but usually are. As to why they're called "Hackney carriages" - the last person to know the reason probably died in 1863.)