Difference between "taxi" and "cab"

There isn't any difference between these two, both act as a vehicle of transport for the passenger to get from one place to another.

Although, cab might be more of an olden-usage, as it can also mean any of various horse-drawn vehicles, as a hansom or brougham, especially one for public hire.

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.

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.