Distinguishing between although and while

Question

What are some circumstances in which "although" and "while" are semantically interchangeable? When are they not?

Answer 1

Here's an example:

While we would like to accommodate your request, we have no vacancies.
Although we would like to accommodate your request, we have no vacancies.

But we cannot simply reverse the sentence with "while" as we can with "although":

OK: We have no vacancies, although we would like to accommodate your request.
NOT OK: We have no vacancies, while we would like to accommodate your request.

The concession clause of that type must introduce the sentence if you wish to use "while".

Yet when a comparison is being set up, *while" introducing the sentence means "on the one hand":

While [on the one hand] the emu is flightless, the hawk flies very swiftly indeed.

and when "while" connects the two clauses, it means "on the other hand":

The hawk flies very swiftly indeed, while [on the other hand] the emu is flightless.

"Although" could be substituted for "while" in both of those sentences.