What's the difference between a mathematical proof and mathematical evidence?

I found a paper that try to explain the distinction:
http://www.math.harvard.edu/~mazur/papers/Math.presentation2.pdf

As par as I can see a Math proof is what we know it to be, an 'ironclad' sequence of Math rules applied, that end up leading to a single and irrefutable answer. It can be hugely complex though, so it can take years before a given proof is finally accepted or rejected.

Evidence is fuzzier. it point in some direction but it is not definite, what we looked at may have been an exception or just a coincidence.
An example in language terms: We know that everybody dies sooner or later and we Know that everybody drink milk as babies. that is evidence pointing towards a conclusion that drinking milk kills people.
In this case the evidence was not valid. just because 2 things always happen in peoples lives does not necessarily mean that one thing is what causes the other. the evidence here is coincidental and invalid. But it is still evidence. Just not evidence to support that particular conclusion. More and better evidence is needed before we really can conclude anything about why people die :) This post was edited by JakobA the unAmerican. at January 2, 2017 12:24 AM MST

A mathematical proof is based on deductive reasoning of an axiom and has been shown to always hold true.   All scenarios and possibilities  have been worked out and has shown to be logically true in all cases.

If there is mathematical evidence then not all possibilities have been worked out or as of yet discovered.  There are still other variables that have yet to be worked out or that some level of mathematical conjecture is assumed for the theory to work.   Since it starts from a conjecture it cannot be considered a a true proof since it's based on an unproven premise.   Mathematical evidence is basically conjecture in and of itself.

Basically one is solely based on what is known to be true and the other is based on some assumptions or has some variables that haven't been worked out yet.

I would say mathematical evidence can be demonstrated with probabilities and statistics whereas a proof can be shown to be true without uncertainty, QED, using equations like the square of a hypotenuse being equal to the sum of the squares of the other two sides. I forgot how to spell pythagorius, pythagreous, pathagorius, whatever...