What are differences between vortex, swirl, eddies and wake?
Wake is the general term for air disturbed by a passing aircraft, as might be guessed from the word's nautical origins.
Vortex (in aviation, usually a wingtip vortex) is a part of the wake that originates at the tip of a surface that has a pressure differential.  It's the preferred term in scholarly writing.
Eddy by now seems to be restricted to the name of a general model for turbulence, https://en.wikipedia.org/wiki/Large_eddy_simulation.
Swirl has a technical meaning that I can't quite fathom, but it seems to apply more to combustion in turbines than to airflow outside the aircraft.