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:
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.
I would translate "a través de" as "through," and "por medio de" as "by means of," or "using." The difference is rather slight.