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.