Difference between "take a taxi" and "get a taxi"

In the US you take a taxi, not get a taxi. If you say you got a taxi, the listener would assume that you bought a taxi. If you want to go by bus, you say that you want to take the bus. Or you get on the bus and get somewhere.

In the realm of travel, to take something is to use it as your mode of transportation:

take (transitive) to use a particular type of transport

take a bus/train/plane/taxi etc: I usually take the bus to work.

In British English, get is more common than take for this usage:

get (transitive) (never passive) to use a particular vehicle to travel somewhere

He usually tries to get the 9.03 train.

In American English, to get a taxi, etc. is to obtain its use. I might ask my hotel concierge to Please get me a car to South Station, or inquire of my host Is it easier to get a cab on Lexington or Park Avenue? Similarly, to get a flight is to reserve a seat on a flight, whereas to actually fly is to take a flight.

There are a variety of other ways to express the same— I can catch my transportation (e.g. catch the Shinjuku Line, catch a flight, catch bus #42), ride in/on it (e.g. we rode the tram back to Palm Springs), or simply go on/in/by something (e.g. Can I reach LAX by Metrorail?) among others. But not all of these may apply to all modes; we fly rather than ride aircraft, for example, even though we are passengers and not pilots most of the time (e.g. We always fly Air New Zealand to Fiji).

The rules are different for taxis than for buses and trains.

To get a taxi is to secure the services of a taxi for transportation. To take a taxi is to ride in one. That is, you have to get a taxi before you can take a taxi:

You'll never get a taxi in this neighborhood/at this time of night/in this weather

We got a taxi at the airport and took it to the hotel.

On the other hand, it's pretty unusual to get a bus or a train (though not impossible). It's more common to take or catch a bus or train. As you can see in this ngram, the forms with "get" for buses and trains are much less common.

In this case take refers to the physical act of going and get to the process of acquiring. So they roughly mean the same thing, but with a different emphasis. I call the taxi service to get (acquire) a taxi. When it comes I will take it (ride in it) to my destination.