'Grab' can be used informally to mean that you want to get something quickly especially if there's an opportunity. It is used the same way as when someone says 'I'll grab a drink with you sometime.' Taxi and cab mean the same thing. It just depends on the person who uses it. :)

The original word was "taxicab," and it got shortened in two different ways, both of which became accepted as words. A "cab" originally meant a one-horse carriage for hire, and in turn it was a nickname for "cabriolet!" A "taximeter" (a device for measuring how much the fare is). When a cab is cruising, we have to signal it by waving our arm to indicate that we want it to pick us up. The usual word for this is "to hail a cab." However, we are sometimes competing with other people who want cabs, and we use our arm and hand to do it, so figuratively it has something of a feeling of "grabbing." To "grab" also has the sense of something done quickly or without planning.

@Jerry, I guess you're right. But most of the examples, I can think of, contain one official word and a slang version. In this example, two versions seem to be acceptable.

Lots of languages have several words for the same thing. Doesn't Ukrainian or Russian?

Hello :) Taxi v Cab in my mind.....'Taxi' is a more general term in the UK anyway, and is likely to be a privately booked car you would order from home, for example. 'Cab' is most likely referring to a local council or local government licensed "black cab" which we have in the UK, or a "yellow cab" like the ones seen in New York. You are more likely to 'hail ' one of these passing by on public roads or 'grab' one of these parked in a rank or bay. It could also be a country thing, for example in the UK you are more likely to use the word 'taxi' and in the US it would more likely be 'cab'. I suppose the phrase 'grab' could be a reference to grabbing the handle of the door, so you would 'hail' a moving taxi in the street, putting your arm out to signal you would like picked up, and then grab the door to open it. Or, at a rank or bay where they wait for customers, you would grab the handle to jump in. Either is good though :) I suppose like you would grab a bite to eat when you are referring to a quick meal. hope this helps.