I’ve been thinking that the fundamental difference is that, in Keynes’s story, people make quantity adjustments, whereas in Hayek’s they make price adjustments.
Does that seem correct?
I tried to answer in the comments, but I think my answer there is incomplete. While I do not think there is a single fundamental difference between the two — instead, there are various differences —, I suggest two major ones. First, Keynes believed that long-run quantity adjustments are not possible, because short-run price adjustments are not possible. Hayek believed that long-run quantity adjustments would come about as a result of short-run and medium-run price adjustments. Second, the two had completely different perceptions of the relationship between consumption and the demand for capital. Essentially, if I were to mark one fundamental difference, related to the two previously listed, it would be that Keynes interpreted the capitalist system to have a fatal flaw: its inability to maintain effective demand.
With regards to the first difference I mention, it not just about price rigidity — where I think my original answer is misleading. Keynes is explicit in arguing that, even with perfect price flexibility, a fall in nominal wages cannot lead to full employment. This is because a fall in nominal wages will lead to a fall in the price of output, decreasing the scope for a possible rise in nominal demand for labor.
The second difference, I think, is fascinating. In The General Theory, effective demand is the sum of two variables: D1 and D2. The first is consumer demand and the latter is investment demand. Keynes saw as a fatal weakness in the capitalist economy that as society becomes wealthier the propensity to consume will fall. This means that D2 will have to rise to compensate for a fall in D1. In his world, though, D2 is limited by two factors. The first is a limit to possible avenues of investment — Keynes thought it was possible to one day satiate consumer desires (a superabundance of sorts). The second is more practical, and it involves the perceived relationship between consumption and investment. If there is a fall in consumption, then there is no incentive for investment: there is nobody willing to consume what is produced.
Hayek saw things differently, largely because he had a completely different conception of the structure of production. He believed that the relationship between consumption and production is indirect. A fall in consumption will lead to a rise in investment. This is not just because — as some “naïve” Austrians like to claim — because an increase in real savings is required to fund investment, but because a fall in consumption changes the relative prices that guide production. Hayek would later call this the Ricardo Effect and presented his idea comprehensively in Profits, Interest, and Investment.
Our two protagonists had two very different views of how the market process works, and it shapes the multitude of differences between their respective beliefs.