One of my first steps to analyze any possible terminology (or problem of words in general) is etymology. This method is not without pitfalls, but some of the age-old assumptions ported along by the words can be illustrative.
The word demonstrate translates vaguely into "to utterly show" (a "monster" is a thing seen—interesting, no?) The prefix de- sometimes being an intensifier.
A "description" is primarily a "write up"—or we could roughly say, a report of something. Usually containing details that help us identify it. Thus, we can see that they both have an overlap in the general area of sense experience. A description that contains no details that a primary experience would reinforce, is rather useless as a "description", and we can demonstrate it's raining by pointing to the sky: "see with your own eyes."
So they have an overlap, and as such, I would guess no easy separability. Would a positivist-to-verificationist accept that you can describe anything that cannot be in some way demonstrated? To part the two, however, "description" would remain closer to the sense of something somebody wrote down on paper, effluvia of somebody's state of brain.
However, I would say that in general there is some irresistible sense that a description is less committal than a demonstration. I can describe rain outside--even if your phone says no sign of rain--but I can't demonstrate it unless I show you it is. If I commit to get you up from your card game (or whatever) to come out to look at the weather, then—if it's not a lame prank—if better bear the debt to settle the matter.
However, I would say that we retain a sense of demonstration that does not require the causes—if my claim is "I will demonstrate rain"--not just demonstrate that it is raining, then yes, obviously what one demonstrates, one has, in most cases, to reproduce, which means that you're trying your hand at the pull levers of phenomena, not simply giving an account of them.
Also, in mathematics, I would expect that instructions to "demonstrate" I would have to prove something, where as in the unfamiliar case of "describe", I wouldn't.
However, still, if I'm a cunning sort I can demonstrate something that I can't necessarily describe in that much detail that would outlast my age. There have been charismatic people who can demonstrate that they can cause other people to do things for them, knowing only an intuitive path from intention to effect. This doesn't make them terribly insightful into the psychology of other humans. The simply know how to pull levers. But they will "demonstrate" feats of human manipulation to their friends as the subject of bets.
Also any primitive can demonstrate that trees will burn, without knowing the details of Lavoisier could have described.
So I go back to etymology: I show you something with "demonstrate"; I simply give you an abstract of an experience with "description", probably as writing became to be known for uninterrupted stream of detail, writing became more apt for the concept of describing (akin to how "logic" seems to come to us from the Greek logos as "word-stuff".)
So I should think that the domain of description is all schools that use words, and the domain of demonstration, carrying with it the implications of empiricism--through the school's own filter of what an empirical experience is.
The reason that I put this qualification on is that some reductionists referring to Benjamin Libet's experiments doubt that we can even demonstrate that we can intend to move our wrists in a circle. In a number of interpretations, our wrists move and we interpret that move through the delusion of intentionality that our beast self demands upon reality--possibly to increase the feeling of power(??). Thus if the beast cannot set up a simple wrist move, the epiphenomenon of intent-to-cause needs not be reliable as well, it simply joins the other epiphenomena we associate with empirical activities.
Overall, however, I'm wondering where the (pardon the use) fixation over these two words came from? I think the value to words is that they mark a set of expectations--a sort of social contract--that we invoke when we use them.