I have known a good number of scientists, both social and physical, and I've never once heard them use this expression non-ironically to describe either their own, or anyone else's work. Mathematics proves things, by formal theorems. Science, on the other hand, provides evidence that supports some hypotheses and which rejects other hypotheses. But even when a null hypothesis is formally rejected, knowledge in the sciences is contingent. At any time, your theory is making falsifiable predictions that are so far consistent with the data, but which might be overturned at any time.
And even in places like economics, theory models, which do use formal mathematical proofs of particular ideas and thus may loosely be justified in terms of speaking of 'proof', almost never use the term when referencing the broad idea they're trying to advance. Economists will say 'I solve a model which shows how information asymmetry affects trading volume', not 'Information asymmetry is scientifically proven to decrease trading volume'. What has been solved is one particular model, but there are many other competing models that may be consistent with the data too. Nobody would dream of saying that science proved their theory result.
'Oh sure', you might say, 'we understand that there's a distinction among the finer points of philosophy of science. But in practice, saying science has proved something just means there's lots of evidence consistent with it. Why be such a purist?'
A good question, since you asked.
The reason my heuristic works, however, is that most people who perform actual science do understand the distinction, and are likely to use the right language. By contrast, people who like the phrase 'scientifically proven' are almost always sneaking in an appeal to authority in order to paper over either a) their lack of understanding of the complexity of the issue, or b) the annoyingly inconclusive evidence for the particular proposition that they think it would be politically desirable for more people to believe.
The claim in the above paragraph, of course, is a hypothesis. In the name of science, we should see whether the evidence supports the hypothesis or not.
To check, here's the top 5 results that come up when I type in the phrase 'reject the null hypothesis' into Google News:
1. Do Teams Undervalue European Skaters in the Draft?
2. Hypothesis Testing in Finance: Concept & Examples
3. Culture war in the deep blue sea: Science’s contentious quest to understand whales and dolphins
4. WaPo Climate Fail on Missouri
5. Using a fund manager? You'd get the same results at a casino
So that may not sound stellar, but they're all somewhat related to formal evaluation of evidence for and against ideas in the social or physical sciences. Now compare it with what comes up for 'scientifically proven':
1. Scientifically proven herbal aphrodisiacs
2. Writing Exercises Scientifically Proven To Redirect Your Life
3. 10 scientifically proven ways love can heal!
4. Emojis Are Now Scientifically Proven To Help You Get Lucky
5. Ryan Gosling’s Face Has Been Scientifically Proven To Make Men More Supportive Of Feminism
In other words, worthless clickbait. Colour me shocked.
The results, while not subjected to formal statistical testing, directionally support the hypothesis that 'scientifically proven' is a brain-dead appeal to authority by lazy English majors who wish to unjustifiably associate their claims with the patina of scientific credibility.