Daniel Sanchez of the von Mises Institute recently wrote an excellent article titled “The Odyssey of Sound Economics” about the history of the Austrian School of Economics from its founding by Carl Menger in 1871 to today. The Austrian School is not a monolithic body of thought. There have been different approaches to various aspects of economics within the Austrian tradition over the years. Sanchez traces the first split to the beginning of the so called “marginalist” revolution of the 1870s due to the pioneering work of Carl Menger, Leon Walras, and William Jevons. I used quotes because subjectivist (meaning subjective valuation that replaced various objective value theories of classical economics) is a much better description of this revolutionary approach to economics. From the start, Menger championed a causal-realist approach as he attempted to explain the formation of prices in the real world, while Walras championed the general equilibrium view (aka the evenly rotating economy) that can be no more than an abstraction as the equilibrium state will never occur in the real world.
The Mengerian branch produced: Eugen von Böhm-Bawerk -> Ludwig von Mises -> Murray Rothbard. The Walrasian branch produced: Friedrich von Wieser -> Joseph Schumpeter -> Friedrich von Hayek -> Lionel Robbins.
This split was in no way complete, meaning that there has been a significant amount of overlap between the two groups. Additionally, there have been splits within the these groups (for instance the concept of entrepreneurship).
Personally, I favor the view of the first branch, that of causal-realism that truly embraces subjective valuation. In fact, today this is what is usually considered Austrian economics. The second branch either ended when Hayek turned away from pure economics after the 1930s or merged with mainstream neoclassical economics.
I encourage readers to read Sanchez’s article. He does an excellent job of showing the historical roots of today’s revival of the Austrian School of Economics.