TITLE: Is a proof also "evidence"?
QUESTION [21 upvotes]: Can I use the terms proof and evidence synonymously or is there a difference? You usually see mathematicians writing about "proof" while other sciences instead discuss "evidence" - is there a practical difference or do they mean the same, an understanding of the basis for the scientific result and it is that basis that we either call proof or evidence?

REPLY [3 votes]: I'd like to add an aspect to your question, that hasn't been touched by any of the other answers thus far:
Evidence plays a significant role in current day mathematics.
Vectornaut already mentioned one example of this. Let me provide another, that has a different kind of flavour to it.
I'd like to talk about large cardinals, but before I do so, let me provide a definition that is suitable for our purposes. Let us say that $\kappa$ is a large cardinal iff there is a set theoretical formula $\phi$ such that 
$$\operatorname{ZFC} + \phi(\kappa) \vdash "\text{there is some weakly inaccessible cardinal } \mu \le \kappa"$$
(Please note that the term large cardinal doesn't have a widespread definition in the literature and there are properties that people (including me) would regard as large cardinal properties but don't satisfy the definition above.)
Using my definition and assuming that $\operatorname{ZFC}$ is consistent, we cannot prove that there is a large cardinal property $\phi$ and a cardinal $\kappa$ such that $\phi(\kappa)$ holds. In fact, the assumption that any large cardinal exists is strictly stronger than the assumption that $\operatorname{ZFC}$ has a model and it seems reasonable to believe that large cardinals have to lead to contradictions. In the past there have been some examples of large cardinal properties that turned out to be inconsistent (see for example Reinhardt cardinals).
However, over the last 40+ years, people intensely studied large cardinals and where able to derive surprising and beautiful theorems from their existence. (For example, the existence of $n$ Woodin cardinals and a measurable above all of them implies that $\bf{\Pi}^{1}_{n+1}$-determinacy holds, for any given $n < \omega$. And conversely, for any given $n < \omega$, $\bf{\Pi}_{n+1}^{1}$ implies the existence of an inner model with $n$ Woodin cardinals.
Since we cannot prove that a given large cardinal assumption is consistent, relative to $\operatorname{ZFC}$, we have to rely on evidence, rather than proof. One way to 'strengthen' this approach is, for a given large cardinal property $\phi$, to study larger large cardinals $\lambda$ such that the existence of $\lambda$ implies the existence of many $\kappa < \lambda$ such that $\phi(\kappa)$ holds. If we can't prove, in this setting, that the existence of $\lambda$ is inconsistent, this is stronger evidence for the consistency of the existence of $\kappa$ such that $\phi(\kappa)$ holds, than just the fact that we weren't able to prove the inconsistency of such $\kappa$. Obviously, this isn't rigorous in the sense that we can be sure that a certain large cardinal assumption is consistent, but the same holds true for $\operatorname{ZFC}$ itself. By pushing its boundaries one merely gathers evidence and - over time - gains some reasonable confidence in its consistency.