When should one consider using GMM? One of the things which makes econometrics unique is the use of the Generalized Method of Moments technique.
What types of problems make GMM more appropriate than other estimation techniques?  What does using GMM buy you in terms of efficiency or reduced bias or more specific parameter estimation?
Conversely, what do you lose by using GMM over MLE, etc.?
 A: The implications of economic theories are often naturally formulated in terms of conditional moment restrictions (see e.g. the original asset pricing application of LP Hansen) which nest a variety of unconditional restrictions thus leading to overidentification. Rather than arbitrarily picking "which squares to minimize" to satisfy a subset of those restriction exactly using whatever-LS, GMM provides a way of efficiently combining all of them.
MLE requires a complete specification - all of the moments of all the random variables included in the model should be matched. If those additional restrictions are satisfied in the population, you are naturally getting a more efficient estimator, perhaps, with a better behaving objective function to be optimized.
In the context of simulation estimation, however, nonlinearity of likelihood functions introduces an additional source of bias, complicating the comparison with SMM.
A: GMM is practically the only estimation method which you can use, when you run into endogeneity problems. Since these are  more or less unique to econometrics, this explains GMM atraction. Note that this applies if you subsume IV methods into GMM, which is perfectly sensible thing to do.
A: One partial answer seems to be that:
"In models for which there are more moment conditions than model parameters, GMM estimation provides a straightforward way to test the specification of the proposed model. This is an important feature that is unique to GMM estimation."
This seems like it would be important but insufficient to wholly explain the popularity of GMM in metrics.
