Solution 1:

The reasoning that appeals to me here is the sense of "entirely" for the prefix "de-". (Acknowledgements to etymonline.com). Privation might be simply me being separated from my cat for a day. Deprivation would imply that my cat died.

Privation of water from beans is not enough for them to be stored well; they should be deprived of water, desiccated, and then they will store.

Common usage of course has blurred the distinction. Can I be deprived of my rights? I can always get them back at some point if lost; but if the temporary separation from the right results in my death, then the result is extreme.

Since the true meaning of de- is lost in time like tears in rain, this is only one of many possible answers.

Solution 2:

From M-W:

deprivation: the state of being deprived

deprived: not having the things that are needed for a good or healthy life

privation: an act or instance of depriving : deprivation / the state of being deprived; especially : lack of what is needed for existence

Based on these definitions, deprivation and privation seem to mean almost the same thing. If there is a difference, it concerns "not having the things that are needed for a good and healthy life" (deprivation) and "lack of what is needed for existence" (privation). From the point of view of life, having to endure privation seems slightly worse than having to endure deprivation.