We use both in the UK. This so-called difference is one of the many myths about GB/AmE English.

In Australian English you would say "taxi" rather than "cab", but either one would be understood easily. ("Cab" sounds American to my ears)

Taxi or Cab in the UK, as far as I know that doesn’t vary much by region. I think in use it depends on how the phrase is spoken - e.g. ‘let’s call a cab’ is just easier to say than ‘let’s call a taxi’, reverse for ‘we can take a taxi/take a cab’, for the obvious reason of c-c and t-t just tripping off the tongue more easily. You might hear an older and well educated person say ‘taxicab’, but it’s not common in current usage.

In the United States, we use both taxi and cab interchangeably all the time, though depending on where you live in the states, one might be a bit more commonly used than the other, but still interchangeable regardless. As far as I know, both words are interchangeable in Canadian, Australian, and British English as well, though as with different regions within the United States, one may be a bit more commonly said depending on which country it is, while still being interchangeable and universally understood regardless of location/type of English being spoken.

In American English, they are interchangeable, though one may be used more than the other depending on where in the states you are. For example, people may use "cab" more in New York, whereas people don't use either in Los Angeles (everyone just uses Uber!)