Impedance matching using transformer vs other circuits, what is the difference
I'd use a transformer when I want to isolate one signal from another. The transformer provides galvanic separation between two circuits and this is needed in several applications. One that springs to mind is a data coupler between a rotating machine and a static data receiver - you can't use wires but, you can use a rotating transformer to couple the data.

I've used near-field data transfer of modulated data as well and, in effect, antennas have become a transformer. 

You can find RS485 chips that now embody isolation transformers and, ADI have a series of data coupling chips that incorporate tiny transformers to provide galvanic isolation.