As far as I know the Vienna Circle was very relevant to science in the twentieth century. Why? What was the importance of the members' philosophy in science? Would science be too different from what it is today if it weren't for them? (I know these kinds of question are quite tricky; a comment is enough)
The Vienna circle, and more broadly logical positivism that it promoted, was the key link connecting philosophy of classical physics in 19-th century, positivism and neo-Kantianism of the Marburg school, to the modern philosophy of science post the discoveries of relativity theory and quantum mechanics. There is a direct line of tradition from Kant, Mach and Cassirer, through Carnap, the most prominent figure of the circle, to Quine, Popper, Kuhn, Feyerabend and Lakatos. Logical positivists were among the first to meet the philosophical challenges presented by the fundamental changes in physics at the turn of the 20-th century, and revise the conceptual framework inherited from their classical predecessors. Friedman's book gives a detailed account of the Vienna circle's ideas, their genesis and interaction with the science of the day.
Perhaps, the most direct influence of the Vienna circle on science proper was in shaping the Copenhagen interpretation of quantum mechanics. Both Bohr and Heisenberg were personally acquainted with the circle members in 1920-s, and incorporated verificationism, one of the central tenets of the logical positivism, into their interpretation. Verificationism asserts that empirically unverifiable statements are meaningless, and questions concerning them are pointless. This was Bohr's main line of defence against Einstein's criticisms of quantum mechanical "incompleteness". Another direct contribution, to formal semantics, came from the related focus of logical positivists on the logical analysis of language.
The development of falsificationism by Popper since 1930s, Quine's criticism of verificationism in Two Dogmas of Empiricism in 1950s, Kuhn's historical analysis of paradigm changes in science in 1960s, etc., undermined the fundamental positions of logical positivists. Although the original doctrines of the Vienna circle in philosophy of science are rarely defended today what came to replace them emerged in no small part from interaction with and revision of their ideas. Carnap himself was instrumental in publishing Kuhn's book about scientific revolutions, and saw the historical aspect of paradigms as an important addition to his own work on analysis of scientific language.
But perhaps the circle exerted its greatest influence by osmosis. It was founded in 1922 by Moritz Schlick and officially functioned until 1936, other prominent figures included aforementioned Carnap, Neurath, Hahn (of the Hahn-Banach theorem) and Menger. However many others, Popper, Quine, Reidemeister, Gödel, Tarski, etc. were either associated or actively interacted with the circle at various times. At the time the circle was almost a lone proponent of a rationalist approach to philosophy and science in Europe increasingly dominated by anti-intellectual trends. After the spread of Nazism several key members of the circle emigrated to the United States, where their ideas found far more favorable reception.
The Vienna Circle consisted of two economists - namely the Menger's (among many other heavily credentialed geniuses). They were influential in the social sciences, namely those that apply game theory. The son even started his investigation with an exploration as to the success of mathematics in explaining economics from a purely rational point of view. That sounds like the beginnings of philosophical questioning to me! After all, one can easily trade out economics for any other field in which game theory can potentially be applied in such a questioning to obtain a philosophical question - namely how much can game theory apply to the world?
But then, it should come as no surprise that when new fields are forged in the sciences there is new history for philosophers to question and contemplate.
"Philosophy of science without history of science is empty; history of science without philosophy of science is blind." -Imre Lakatos