It seems like a través de and por medio de can often be used to mean the same thing. What is the difference between them, and in what contexts can you only use one or the other?
There's little (if any) semantic difference between these two constructs, as they both can express mode:
Hicieron el pago a través de transferencia bancaria.
Hicieron el pago por medio de transferencia bancaria.
Se hizo paso a través del paseo marítimo.
Se hizo paso por medio del paseo marítimo.
However in the first pair of examples, por medio sounds awkwardly more formal than a través, and in the second one, rather uncultured (if that's the right word in English).
There are cases in which por medio sounds just as acceptable as other options, but as a rule of thumb, always prefer a través de.