Theoretical Applications for Approximation Algorithms Lately I've started looking into approximation algorithms for NP-hard problems and I was wondering about the theoretical reasons for studying them. (The question is not meant to be inflammatory - I'm merely curious).

Some truly beautiful theory has come out of the study of approximation algorithms - the connection between the PCP theorem and hardness of approximation, the UGC conjecture, the Goeman-Williamson approximation algorithm, etc.

I was wondering though about the point of studying approximation algorithms for problems like Traveling Salesman, Asymmetric Traveling Salesman and other variants, various problems in mechanism design (for instance in combinatorial auctions), etc.
Have such approximation algorithms been useful in other parts of theory in the past or are they studied purely for their own sake?

Note: I'm not asking about any practical applications since as far as I know, in the real world, it's mostly heuristics that are applied rather than approximation algorithms and the heuristics are rarely informed by any insight gained by studying the approximation algorithms for the problem.
I strongly disagree with the last paragraph. Blanket statements like that are not useful. If you look at papers in many systems areas such as networking, databases, AI and so on you will see that plenty of approximation algorithms are used in practice. There are some problems for which one desires very accurate answers; for example say an airline interesting in optimizing its fleet scheduling. In such cases people use various heuristics that take substantial computational time but get better results than a generic approximation algorithm can give. 

Now for some theoretical reasons for studying approximation algorithms. First, what explains the fact that knapsack is very easy in practice while graph coloring is quite hard? Both are NP-Hard and poly-time reducible to each other. Second, by studying approximation algorithms for special cases of a problem one can pin-point what classes of instances are likely to be easy or hard. For example we know that many problems admit a PTAS in planar and minor-free graphs while they are much harder in arbitrary general graphs. The idea of approximation pervades modern algorithm design. For example, people use data streaming algorithms and without the approximation lens is hard to understand/design algorithms because even simple problems cannot be solved exactly.