The difference SHOULD be that "proof" means you can close your mind, while "low probability" means that you can discount it but you still have to retain a certain openness to the possibility.

Far too many things have been considered low probability simply due to the inability of the current science or imagination to grasp their possibility. Then, when they happen, too much time is wasted while people argue that the negative was already somehow proven.

I submit for your consideration both Global Warming and Quantum Mechanics.


In sum, I use "You can't prove a negative" to try to remind people that the opposite is always possible, no matter how improbable, unless a contradictory truth has been proven. If nothing else, one hopes it prevents smugness.