Now, "proof" is a dangerous word in science. It's vague, it's literally impossible to do in a truly strict sense within the scientific method, and people are quick to take the term to mean something it doesn't.
It can mean anything from basic empirical evidence of a contentious event occurring(like proof that flies lay eggs) to a theoretical framework so soundly tested and retested as to lack detectible flaws(like the law of attraction).
The philosophical or mathematical proof takes premises, and using absolute rules arrives at inescapable and undeniable conclusions. In a sense, this is possible with science: Assume an object is in motion at a certain velocity v. Assume it's position is x. Assume no force is applied. After t time, inescapably it's new position is x+v*t, by deduction. But science allows that to be wrong just as soon as someone comes forward with an experiment where it doesn't happen(though our first guess would be that one of the assumptions is untrue, given just how reliable laws of motion are). You can never have a proof that is just true.
Quick to confusion:
The various definitions here are easy for people to conflate or mistake. Just look at people expecting "proof" of evolution. They simultaneously want empirical evidence of a contentious hypothesis, like you'd test in a lab, and applying the concept to an entire branch of study, which has an entirely different idea of "proof". It all adds up to a scenario where people don't get what it is that they don't get.
And they have very high expectations from pop culture: person in lab coat gets item, "aha we know what this is now that we've run 'tests'". They see science as much the same.