I have several responses to this; I'll start with criticisms:
First, the article is really only about macroeconomists. They are important, and a big chunk of the profession, but hardly all of it. Talking about economists as a vague whole misleads the reader. There is no problem along these lines in development economics, for example, or in the economics of education. Indeed, if anything, those fields would be better balanced if they had a bit more detached theory.
Second, even within macroeconomics, Thoma surely overstates the problem. At least one of my macro colleagues shuttles back and forth to DC all the time to provide aid and counsel on policy choices. A look at any macro journal will reveal a mix of theory and empirical work, with much of the empirical work devoted to estimating policy relevant parameters.
Third, I think a bit more recognition that there is a tension between specialization and division of labor on the one hand and lots of interaction between academics and practitioners would have improved the piece. I suspect that the optimal setup includes some academics who specialize theory and some who delve more into the real world of application. What's that you say ... that is what we have now? Well, perhaps.
Fourth, I think Thoma overstates the value of forecasting. Academic economists disdain forecasters not because they use old models, but because they are mostly selling snake oil, which is to say that most forecasts are not very good. Indeed, as John Cochrane has pointed out many times, in some sense the model says that forecasts should never be very good because current prices already reflect all the available information. In this sense, improving the knowledge base - they "how it works" knowledge that Thoma criticizes in his piece - may be the best way to improve the forecasts that matter most, which are the ones implicit in current prices.
Fourth, I am not convinced that science = "theory and math".
At the same time, I do think there is something to what Thoma has to say. Part of why I do some consulting is precisely to have the interactions with practitioners that Thoma describes. I have learned a huge amount from these interactions about how the econometric methods that I study and use in my academic work get understood and applied in real world applications. Some of those lessons have improved my later academic work and/or guided my choice of things to work on.
I have also seen academic economists who feigned knowledge of particular applied areas but in fact had no clue about them. In my experience this is most common in industrial organization. That field has experienced a methodological revolution in the past 10-15 years. One result has been a strong emphasis on tools, particularly among students on the job market. Improving tools is a very good thing, but for the moment one of the old virtues of the field, which was strong subject area knowledge, has fallen a bit to the wayside. When I was at UWO we hired an applied theorist who claimed their job market paper showed that Microsoft should be broken up by the anti-trust authorities. In fact, the model in the paper had nothing really to do with the Microsoft situation; rather, some misbehaving member of the student's dissertation committee had pushed them to claim that it did so that they would appear more "applied" and topical on the job market.
So, yes, there is a point here, but at the same time we don't really need to have evolutionary biologists hanging out in operating theaters, to borrow Thoma's medical analogy.