To hail a taxi' means exactly to beckon or to do a call signal for the taxi to stop and for you to use the public service afterwards, whereas 'to catch a taxi' refers to the whole action of making a taxi stop and using the service.
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synonyms: cab fare. type of: fare, transportation. the sum charged for riding in a public conveyance.
A taxi, also known as a taxicab or simply a cab, is a type of vehicle for hire with a driver, used by a single passenger or small group of passengers, often for a non-shared ride. A taxicab conveys passengers between locations of their choice.