Difference between "taxi" and "cab"

There may be a regulatory distinction. In at least some parts of the UK, a taxi can be hailed at the side of the street, while a minicab or more properly "private hire vehicle" must be booked (e.g. by phone). Cab can then be used for either of these. Taxis are subject to more stringent licensing. It is however variable, London has different rules, and they have evolved, terminology may not have kept up with the rules.

read this somewhere.. Found it interesting so thought of sharing here...

Taxi VS Cab

The difference is lost in the mists of time. The earliest form of horse-drawn vehicle available for hire was called a 'cab' (short for cabriolet). The name stuck when cab firms upgraded to motorized vehicles, fitted with a 'taximeter' (which measured how far you'd gone). These were called 'taxi-cabs'. Nowadays either word is used.

In UK the word 'taxi' is used for the diesel-engine ones with a high roof to the passenger compartment (also known as 'black cabs/black taxis'), and the term 'minicab' is used for standard passenger saloon cars that just happen to be available for hire.

Since the definitions don't show many differences, is it okay to assume that there is no difference between taxi and cab?

Yes, they words that are used to mean the same thing.

Regarding the word "taxicab", this is not used by all English speakers. It seems to be more common in certain parts of the world and not in others. It is not, normally used in the UK.

In Trinidad & Tobago the word cab is almost never used by the local population. Taxi generally refers to any vehicle specially licensed and insured to transport a fixed number of paying passengers at any given time. These vehicles license plate usually begins with the letter H followed representing it is a vehicle licensed for hire and at present two more letters and four digits which will indicate the age or more accurately the time the vehicle was registered by the licensing authority (HDC 2549 example of a typical taxi number) in T&T usually range from a five seater to one that can accommodate twenty five passengers at one time. one can hire a taxi to serve his personal needs or can stand at the side of the road and stop a taxi going in his direction. At the Airports there are specially registered taxis for hire and they do not work the road like the normal taxis but when hired they are not allowed to take any more passengers along their route between the airport and their final destination

Today, in practice, the two terms have blurred, because nearly all "cabs" are "taxis," which is short for "taxicabs." The two separate terms exist because originally, there were horse-drawn "cabs" for hire. The problem was that cabmen were notoriously dishonest, and cheated customers by claiming longer or shorter distances traveled, or extorting money before leaving them in the right place. Then, someone invented "taximeters" - devices that could calculate the correct fare, based on time and distance; the fare would accumulate based on a clock when stopped - by distance traveled when moving. Cabs with taximeters were called "taximeter cabs." Eventually, automobiles were invented. Some automobiles were used as "cabs" - for hire, but without a taximeter. In 1907, when taximeters on automobiles finally took hold in London and New York, they called them "taxicabs" - short for "taximeter cabs;" and also "taxi" - short for taxicab. The distinction persists, to some extent, because hiring a "towncar" or something, paid by the hour or the trip, does not constitute a "taxi" - technically - because it does not have a device hooked up to the engine and a clock, that constantly calculates accumulated fare by time and distance. Uber runs on a new business model, and it is not clear yet what those types of cars will be called in the future.

For some good background on the history of taxis - see: Taximeter, Taximeter Uber Alles - a History of Taxicabs.