To operate as a taxi driver both the driver and vehicle need to be licensed.  You can be licensed as a hackney carriage driver, who drives licenced hackney carriage.  A licenced private hire driver drives a licenced private hire vehicle. 

What is the difference between hackney carriage and private hire

As a hackney carriage driver you can make yourself available for immediate hire on a designated hackney carriage rank, and you can also be waved down whilst the vehicle is being driven.  As a hackney carriage driver you can also carry pre-booked customers.

As a private hire driver, you must operate through a private hire operator and can only carry pre-booked customers. You can't make yourself available for immediate hire in the way that hackney carriage vehicles can.