Historically, why has it been important to have a clear distinction between science and other forms of knowledge? The only thing I have come across so far is the debate between evolution and creation, with Popper's idea of scientific theories being falsifiable being used to classify creation as non-scientific.
Warning: the following answer simplifies the history of the philosophy for brevity.
In 2019, when "science says X" has all but uncontroversially meant X for a long time, it's easy to misunderstand the historical interest in the demarcation problem. It wasn't about saying, "this is science so is right, while that isn't so it's wrong". It was about understanding why science is trustable. (Do you think you know why it is? Maybe you do, but knowledge always has a history.) If you believe science is inductive, as David Hume did, you can't trust science unless you trust induction (which is harder to do, once you've read him). Meanwhile, if before 1900 you think unscientific claims are less reliable - in other words, if you want to make metaphysics (especially theology) the dirty word it gradually became for many - you need a rough idea of what makes it different, or you can't complain about certain ideas not being up to par.
Most post-Humean philosophers who commented on science at least somewhat trusted it, if only because of its transformational effect on the era they were living in. But how did they justify it? Mostly, by trusting/defending inductivism. Popper is of historical significance because he took a different approach of saying Hume was right about induction being unworkable, but wrong about science being inductive in the first place! But of course, if you say that, you have to say what you think characterises science instead.
So what is science? Popper gave his opinion; Kuhn gave his; Lakatos gave his; Feyerabend said it was a mug's game to attempt demarcation; Thagard nonetheless gave yet another view. From Hume 1748 to Popper 1934, science was thought of as inductive. In the decades following that, multiple philosophers prominently gave yet another take on demarcation. Not only did the pace of new proposals accelerate; each new idea led to important new discussions in philosophy about what you can trust in science. For example, if you agree with Kuhn, you'll think modern scientific explanations are incommensurable with older ones, which doesn't bode well for the concept of scientific progress. Similarly, Feyerabend's attack on method is also an attack on scientific exceptionalism, whether you think it deletes scientific knowledge or puts dancing angels back on pinheads. So obviously, whom you agree with matters a lot.
In the US, we might say that "science is the kind of knowledge that we should all agree upon." That's why it's taught in schools, while we don't teach religion: public schools must refrain from contradicting individual freedom of religion.
I tend to think that Popper's criterion doesn't classify just creation as non-scientific, but all explanations of origin, which no experiment could ever be devised to falsify.
The demarcation problem began with Parmenides who distinguished discursive knowledge based on truth orientated procedures from mere opinion. This is the original sense of the notion of the term science. It meant a body of knowledge butressed by rules of evidence, reasoning and justification which is peculiar to the field in question. In this sense, history and politics are sciences and its why in Islam there is the term, the science of hadith or the Qu'ranic sciences - which sounds odd for anyone brought up on a diet of modern science (as it did to me once).
The Parmenidian thesis was then taken up by Plato. For example, his dialogue, Georgias, has Socrates arguing against rhetoricians or as he called them, sophists and who traded in opinion said well - though he admits the utility of things said well and again in his late dialogue, The Sophist. Then through Plato it fed into the broader stream of Western philosophical thinking.
This tradition was inverted by Popper who declared there was no truth procedures but only procedures that can be falsified. This position has become popular in the modern analytical philosophy if science.
Personally, I prefer Parmenides.