These terms seem to have the same meanings. Can anyone differentiate them?
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.
I don't know what a swirl is, but here are the other terms.
Wake is the fluid flow behind a solid body or surface. It may contain vortices, eddies, or none of them (such as behind a non-lifting body or surface in an inviscid flow).
Formally speaking, a vortex is a region of fluid flow that induces other fluid elements to circulate around it; the mathematical term for this "ability" is called vorticity. Worth pointing out is that the vortex may be really thin and confined in size. For example, in the inviscid theory, we often model the horseshoe vortex, which is actually infinitesimally thin; however, this thin vortex line will actually induce a rather large radius of fluid to circulate around it, even though they, by themselves, are irrotational.
In the wake of of a lifting surface, you'd inevitably find vortices. Even if a body or surface is non-lifting, vortices will also be present in the wake due to the shearing effect of friction, which causes fluid elements to rotate and generate vorticity, unless we are in an ideal inviscid flow.
An eddy is a recirculating flow that is turbulent, i.e. time-varying. To sustain the recirculation, there is inevitably vorticity within the eddy. You'd find eddies in the wake of a stalled lifting surface, or in the wake of a bluff body.