The public must be protected from rogue operators when they want a car to pick them up. Taxis advertise that they are taxis and advertise when they are available for hire. Private hire vehicles must be booked in advance via a registered operator, carry only limited advertising and cannot be hailed or tout for business. Both types of businesses are controlled by a local authority to ensure that a safe service is provided to the public. Both operators have specific insurance covering the passengers, the vehicles must be maintained to a high standard and the drivers must perform a good service. Some operators are both private hire and taxi companies.
Technology is blurring the difference between taxis and private hire. In the past a telephone call from a venue or a phone box could be used to call a private hire car or a taxi. A taxi could also be hailed but it would be illegal for a private hire car to stop if hailed. Smart phone aps using GPS linked to a data base of GPS enabled private cars allow these companies step into the shoes of the taxis. Some aps even allow the customers to rate the driver in the same way that Check-a-trade rates services.
I would be happy for either to take me to give me a lift. Does the public care? I think not.