Practically speaking, I would say there is no difference. However, I believe there is a very subtle difference.
"Can I talk to the manager?"
"No, I'm afraid that's not possible." (She's busy at the moment.)
"No, I'm afraid that's impossible." (She's dead.)
Because the stores are all closed, it's not currently possible to go shopping.
Because the stores are all closed, it's currently impossible to go shopping.
I would argue that the phrase currently impossible, while technically correct, is not one that is as commonly used.
To me, impossible, at least in this sense, implies more of a permanence than does not possible. With not possible, the situation may be permanent; but with impossible, it's less likely to be temporary.
So (again, to me), not possible means an arbitrary set of circumstances whereby something just happens to not be the case at the moment. Meanwhile, impossible means a categorical situation that can never be met (1+1=3).
However, that very subtle and debatable difference aside (and I've certainly heard people say "that's impossible" in response to a temporary situation when being asked for someone or something), they are interchangeable.