The Real Problem With Non-Austrian Economics
First of all, it bases the case for praxeology on Kantian epistemology. A better case can be made using Aristotelian/Objectivist epistemology, like Murray Rothbard did, and like I did in my University essay in Theoretical philosophy(not online).
Secondly, in my experience, neoclassical micro economic theory or New Classical or New Keynesian macro economic theory is not particularly empirical. They are at least as aprioristic as praxeology, and in one sense they are in fact even more aprioristic.
While Kantian and Aristotelian/Objectivist Austrians will argue over whether the origin of the praxeological axioms can be viewed as empirical or not, they are certainly empirical in the sense that we do meet them in applied form in our daily experience. I for example act daily, face a scarcity of time and other resources, and face a uncertain future, and thus face opportunity costs in every action I take, but ultimately choose one action based on my assessment of what will be most beneficial for me, and so on.
By contrast, neoclassical micro economic theory is to a large extent completely disconnected from the empirical reality that real people face. Instead, it ludicrously asserts that economic action can be described as functions of differential equations and various other advanced mathematical functions such as Lagrange multipliers and matrix algebra.
These advanced mathematical functions have absolutely nothing to do with the actual economic reality we see, and so will at best once they've been translated again into English (or other verbal languages, such as Swedish) simply come to the same conclusions that we already knew before we performed these equations. Often however, these mathematical equations are worse than useless in enhancing economic understanding, they in fact necessitate the adoption of a false description of reality. One example of this is the focus on equilibrium (a focus necessitated by the fact that the first order condition of a Lagrange multiplier is that the partial derivative is zero, which in this context translated into English means that no further economic gains can be made, or in other words that equilibrium is reached), which in turn means that there are no room for one of the key elements of economic reality, namely entrepreneurship.
Thus, while the praxeology of Austrian economics means a aprioristic understanding of actual economic laws, the mathematics of neoclassical economics means a aprioristic understanding of....well, advanced mathematics. But it does not in any way help you gain a deeper understanding of the subject matter economics is supposed to be about, namely the causal relations of real world economic life.
More or less the same thing is true for New Classical macro economics (including Real Business Cycle models) and to a only somewhat lesser extent also New Keynesian macro models.
It is true that non-Austrian economists usually, apart from having substituted real economic theories for mathematics, also do empirical research, in the form of econometric regression analysis. While I completely reject the usefulness of things like Lagrange multipliers in understanding economic reality, I am actually contrary to many other Austrians, open to a limited degree of usefulness of econometrics. The reason is that while praxeology helps us understand the causal connections of real life economic experience, it does not say anything about how applicable different applied phenomena's are in today's economic reality. And as the purpose of economics is to help us understand the economic reality we actually face, not just the economic reality we could conceivably face, one must analyze the actual data to see how applicable various causal connections. And econometrics can have a limited degree of usefulness in that task, as long as one is aware of the limitations of econometrics. Namely that first of all, an econometric study is just a local study of a particular data series, not necessarily applicable to other data. For example, during one period of time changes in interest rates may have had a very large impact on investment activity, but during another period of time it will only have a very small impact. Similarly, while a certain causal factor may be very important in one country, it may be of lesser importance in another country. And secondly, one have to take even this local result with a grain of salt, as it is impossible in practice to statistically control for all other conceivable causal factors. Understanding these limitations, econometrics can have a limited usefulness in understanding the relative importance of different causal factors. However, if one does not understand these limitations and try to apply econometric results in radically different contexts then the ones where they were derived, then it could reduce, and not enhance, our understanding of the economic reality we face.
But the point here is that in the distinction of aprioristic versus empirical, Austrian and non-Austrians do not differ in any radical way. Both rely on both aprioristic theory while also performing empirical analysis of actual economic events. The main problem with non-Austrian economics, is that they have substituted actual economic causal relations for mathematical functions, a problem which is greatest in the aprioristic version of non-Austrian economics.