To start with, there are two personal incidents.
When I was a graduate student at UIUC, I had a Pakistani friend, Zahid Anwar. He narrated to me that during his undergraduate studies back in Pakistan, once he did an internship in electricity department. He was made supervisor of a lineman. He would accompany the lineman across the city as the lineman travelled from place to place to fix broken transformers.
Now, Zahid knew basics of electricity, like Ohm's law and Kirchoff's law and the like, but fixing transformers was beyond him. The lineman was experienced: he knew where to test what and how to debug what part of transformer needed fixing and how to fix it. Zahid started thinking why Zahid was the lineman's supervisor rather than the other way round.
Then, once while returning back from fixing a transformer, the lineman asked Zahid: "Zahid bhai, do you know what is electricity?" The question startled Zahid. Zahid asked the lineman: "Do you know what is an electron?". The line replied in negative. Then Zahid asked him: "Do you know what is an atom?". The lineman did not know that either. Zahid was at loss as to how to explain electricity to the lineman. Finally, Zahid explained to him thus: "Electricity is like a wind. When that wind blows, there is electricty".
And then Zahid thought, perhaps this is why I am his supervisor. While the lineman knew all about fixing electricity flow in transformers, he did not know what is electricity. And that limited the set of tasks that he could do.
Due to grasp of the first principles, while Zahid could contribute in power plant design, the lineman could only mechanically solve some pre determined set of problems. This illustrates the power of first principles thinking.
The second story is from 2005 when I got my first job. I had two job offers: from Google and VMware. Both were hot startups though Google was more widely known and had higher prestige. In choosing between the two offers I applied first principles thinking. I reasoned that Google made money from ads, and in 2005 ads were not that big, and personally did not remember clicking any ads. So I argued that Google would eventually run out of business and to be profitable they would need to make search paid. That would be the fall of Google as people would migrate to Yahoo which would be free though the results were of lower quality. VMware, on the other hand, made server software that helped enterprises reduce their infrastructure cost. That was a clear business, and VMware was destined for success. I joined VMware.
There were other reasons behind choosing VMware, like the interview experience and my desire to work closer to hardware, but above first principles thinking was there too. And in retrospect the result of that thinking was incorrect. What I needed was second order thinking. I could have reasoned that several smart people are bullish about Google. I could also have done a more thorough job in the first principle thinking - Google was a public company by then, and so looking up the growth of their ad revenue would have been possible.
So, the problem of first principle thinking is that while by definition it cannot be incorrect, you may not be the competent person to do the first principle thinking. When faced with a decision in that scenario, you have to resort to second order thinking.
While the above stories give situations where each type of thinking shone, in most situations, the two types of thinkings are intermixed with each other. However, before considering the intermixing, let's see what thinking shines in which scenarios.
I believe first principles thinking shines in those situations where problem domain is well defined. For instance, in solving maths problems, or in writing programs. In such a situation, a more logical thinker will be able to make better progress than someone who relies on analogy, or second order thinking. Even doing science is a first principles activity. Thus great scientists and mathematicians are great first principle thinkers.
Second order thinking rules in ambiguous scenarioes like business and politics. In these areas one cannot be confident that one is doing first principles thinking correctly, since one's domain knowledge is not complete. Thus, people rely on other people's knowledge (since different people know different aspects of the problem) and gut feel. A business leader needs to have a good second order thinking since the leader would not be able to get deep into each of the different areas that the company needs to excel at (like product development, sales, partnerships, human resources etc), but still needs to make decisions about these various areas.
In real life, we combine the two types of thinkings. For instance, while Mathematicians are first principles thinkers, note how Terrence Tao, the famous mathematicians chooses which problems to solve. He does not arbitrarily picks up problems to solve -- he picks those problems which he thinks can be practically solved. That is the second order thinking.
I believe that first principles thinking is more difficult to learn. It might require correct genes, and definitely needs to be inculcated from childhood. Second order thinking, on the other hand, can be learnt even in adulthood, and life experiences certainly make you better at it.
Thus, while you may see first principle people (programmars, mathematicians) sometime cross over to second order thinking areas (like business) and achieve success, the reverse does not occur. A really fortutious case is that of people who are great in both first principles and second order thinking -- some of them achieve great success in business.
My personal strong point is first principle thinking and weakness is second order thinking. I already gave the job offer evaluation example above. In another instance, when I was making my first web app, it did not occur to me that I should use a database to store customer data (despite having taken a database course). Rather, I implemented a mini database myself. I stored customer details in a file and modified each time a user got added, deleted or modified. While Don Knuth wrote Tex by himself and Linux Torvalds wrote Linux, I did not get too far in writing my own database, but such an exercise indeed helped me understand the difficutly involved in writing a database and helped me understand databases better.
Over time, I have become aware of my focus on first principles thinking so I try not to over apply it. For instance, when working on my startup QGraph, I was working with my cofounder Sachin, the idea of mobile marketing automation was Sachin's and I did not fully appreciate why this should be a useful problem to solve. However, Sachin had run a revenue generating company (albeit low tech) earlier and so that I relied on him that the idea must be saleable. It so turned out that we had a successful company.
In closing we can view problem solving along this dimension of first principles and second order thinking. It is good evaluate what your natural tendencies are be aware of the pitfalls associated with them.