Evidence is some fact that supports a theory. There may also be other, incompatible theories that would also be supported by the same evidence, so either of them may still be right or wrong.
Once there is proof, that means that there is enough of the right evidence that it all supports the theory, and it's not possible that it's not true.
Maybe you believe that Earth has the shape of a cylinder. You travel all the way around it in a ship, and end up at the same spot! Evidence for the theory that Earth is a cylinder, because if it's a cylinder, then that would be possible.
But it's also evidence for the theory that it is a sphere. But then people find the poles, they travel around it in several different directions, and eventually we go into space -- lots of evidence against the cylinder, in fact enough to prove that it's not a cylinder at all (this is just an example, I'm quite sure it was easy to prove Earth is not a cylinder much earlier :-)).
Eventually we gather so much data with satellites et cetera that we know exactly what the shape of the earth is, and it's almost a sphere, but slightly flattened. We know so much about it that that can be considered proven.