In a nutshell, Popper shifted the focus from the justification of beliefs to the formation of critical preferences for publicly articulated (objective) theories.
Care to offer a defense of critical preferences from my criticism?
There are in this orbit no such things as experimentally established facts. All experience in this field is, as must be repeated again and again, historical experience, that is, experience of complex phenomena. [This is Mises speaking.]
I agree with Mises that experimental tests cannot be the focal point of thought about human action. I think he gives a good reason. The data is complex. That means interpretation dominates and the data is secondary. We also, in general, can’t do relevant repeatable or controlled tests.
Tests have limited use in science too. As David Deutsch explained in _The Fabric of Reality_ (e.g. p 66), most bad scientific theories are rejected for being bad explanations, not by doing a test. Tests should be thought of as a very useful way to help settle some of the more difficult scientific disagreements, not as the primary method of science. Seen this way — as more of a bonus than a necessity — it’s not so worrying to do without them in an area such as human action or philosophy.
So Mises was mistaken that his field needs different methods than science in any important sense, since tests aren’t so crucial. Fundamentally, all creating of knowledge is done by the method of conjecture and refutation. No other method has ever been thought of. And supporting epistemological ideas, such as fallibility, rational attitudes to disagreement, the search for good explanations, and the need for error correction, apply just as much to Mises’ work as to scientific work.
Why did Popper think that demarcation was one of the two fundamental problems in the philosophy of science?
He didn’t. People have overemphasized what he said about it, perhaps because it’s one of the few things he said that they understood. Popper explained why he considered it an important problem and why he thought about it in, IIRC, C&R. It’s because he was thinking about Marx, Freud, and Adler and wanted to criticize their claims to scientific legitimacy.