Evidence consists of observations and data that suggest something MIGHT BE true.

Proof consists of data, observations, and calculations that establish that something IS true.

That's just my short form version. You observe something and postulate possible causes. Then you perform studies to determine whether your theory is true or not. The resulting data may show proof that your theory is true or false. If you don't have data that proves your theory true or false, you may design and perform a new study to establish proof.

First of all, if we are talking about science, in science we don't "prove" things. That is a common misconception. Proof happens in mathematics and formal logic, not in the scientific method. In science we use evidence to support or refute a hypothesis. The end result is a theory, which is as good as it gets in science. So, if someone says that they have evidence that proves a theory, then recognize that they are using the language of science loosely.

In science we can say that there is evidence to support an idea. When there is enough supporting evidence then the idea graduates to theory. We can also use evidence to refute ( or disprove, if you will ) an idea. We don't prove ideas, because that implies that it is not ever questioned again. Everything in science is up for re-examination, at all times.

Facts or physical signs that help to prove something.

Evidence of:
Doctors found no evidence of infection.

Evidence that:
There is some evidence that the economy is improving.

Evidence for/in support of/in favour of:
Evidence for the existence of life on other planets.

Proof

Information or evidence that shows that something is definitely true or definitely exists.

Proof of:
We were unable to establish proof of her innocence.

Proof that:
Do you have any proof that this is true?

Source: Macmillan Dictionary.
Proof.
Evidence.

I'm not very familiar with scientific terms. I found out that "evidences" and "proofs" are different, but the explanations were not very clear.
Can someone explain what's the difference between these two words?