Many of the debates between Keynes, Hayek, and other economists working in the UK at the time were carried out in the pages of economics journals, in particular Economica. Hearing these debates summarized and quoted in Wapshott's book, I was immediately struck by how similar they were to today's econ blog debates. There was a huge amount of people talking past each other, quibbling over definitions, and claiming that they'd been misinterpreted. For example, the incredibly byzantine debate between Hayek and Piero Sraffa put me in mind of the Sumner/Wren-Lewis blog debate from early 2012. The sarcastic, bombastic clash of tempers between Keynes, Hayek, and others like Lionel Robbins and Joan Robinson reminded me distinctly of the recent rhetorical fireworks between Paul Krugman, John Cochrane, and others (with Arthur Pigou playing the part of Tyler Cowen, chiding the other debaters for their incivility while advancing conservative ideas in a genteel fashion).
There are some major differences, of course, between the discourses of then and now. The fiery econ debates of the early 20th century advanced a lot more new ideas, and employed more rigorous analysis, than today's blog debates. Steve Williamson often says that you won't learn any cutting-edge economics from reading blogs; but for the back-and forth debates of the 1930s, that was definitely not the case. These days, of course, ground-breaking theories and rigorous analysis have remained in academic journals, and have taken on a far more detached, scientific tone, while sharp debates have mostly moved to the blogs.
As for Keynes and Hayek, the book taught me much more about the latter than the former. Hayek didn't start out as the Anti-Keynes, it turns out - he was a ringer, brought in by Lionel Robbins, William Beveridge, and other British conservatives to battle Keynes when Robbins & co. felt that they themselves weren't up to the task. They brought in Hayek when their first choice, Ludwig von Mises, proved to be too bad at English (and too downright grumpy).
It seems to me that Hayek took his appointed role as the Anti-Keynes a bit too seriously. During his attempt to formulate alternatives to Keynes' General Theory, Hayek repeatedly flirted with ideas that were - in my opinion - much deeper than the simple theory of aggregates that was being advanced by Keynes. These included the mutual inconsistency of economic plans, and the potential failure of economies to reach stable equilibria. Maybe Hayek bit off more than he could intellectually chew, or maybe he shied away from exploring ideas whose implications ran so counter to the political views of his mentor, von Mises. But in any case, Hayek didn't follow up on his tentative forays into the economics of complexity, instability, and disequilibrium, and instead went into political philosophy.
Hayek's most famous piece of political philosophy was, as we now know, completely wrong. In The Road to Serfdom, Hayek claimed that Keynesian-style macroeconomic management would lead to totalitarianism; in reality, nothing of the sort has ever happened. America, Europe, Japan, Korea, and others became solidly Keynesian after World War 2, and while macroeconomic management didn't always work as advertised, it nowhere and never led to the advent of totalitarian regimes.
It's also interesting that Hayek, despite hating the Nazis and totalitarianism in general, seems to have been somewhat influenced by many of the early 20th century Central European ideas that led to the rise of Nazism itself. For example, he repeatedly asserts that people are not created equal, making reference to "superior people," and stating that he would prefer an economically libertarian dictator to a democratic government that restricted economic freedom. This foreshadowed the unfortunate libertarian support for dictators like Augusto Pinochet, as well as more recent libertarian flirtations with "scientific racist" ideas.
Still, the author of Keynes Hayek credits Hayek with keeping the flame of libertarianism alive through the dark winter of Keynesian dominance, and views the emergence of Milton Friedman as the vindication and apotheosis of Hayek's ideas. In this, I feel that the author makes a major mistake; Friedman's monetarism was a type of macroeconomic management that was more amenable to conservatives of the 1970s and 80s than the type advocated by Keynes, but it was central planning nevertheless (as von Mises pointed out). It was Robert Lucas and Edward Prescott who truly restored Hayekian "classical" economics to dominance in the macro field, with their models of frictionless economies and near-optimal business cycles.
In any case, I definitely recommend Keynes Hayek, especially the audio version.