Is it better to use a MLE or a MME to build an asymptotic confidence interval for a real parameter $\theta$? I thought that the answer was pretty straightforward given that the MLEs possess some strong asymptotic properties, i.e. normality, efficiency and consistency.
But then, I have found that also MME (Methods-of-Moments estimators) can be used to build asymptotic confidence intervals, since they are asymptotically consistent for $\theta$ and their asymptotic distribution can be found using Central Limit Theorem + Delta method. 
Thus, how should I answer this question? Are MLEs always better than MMEs or there are some particular cases? 
 A: One criticism of method of moments estimators is that they often aren't functions of sufficient statistics, which means they involve discarding useful information from a sample when estimating an unknown parameter.
Take for instance $X_1, \ldots , X_n \sim$ uniform$[0, \theta]$ and the goal is estimation of $\theta$.  Since $\text{E}(X_i) = \theta / 2$ a valid method of moments estimator is $\hat{\theta} \equiv 2 \bar{X}$.  But $\bar{X}$ is not a sufficient statistic, and as a result our estimator can actually yield impossible values for $\theta$ with a sample such as $\{1, 1, 7 \}$.  Maximum likelihood estimators on the other hand will be functions of sufficient statistics, and you don't end up with situations like this one.  Also, estimators based on sufficient statistics are in general more efficient than those which are not.  To see this let $T$ be any estimator of some parameter $\theta$ and $S$ a sufficient statistic for $\theta$.  Then
\begin{align*}
\text{Var}(T) &= \text{Var}[\text{E}(T \mid S)] + \text{E}[\text{Var}(T \mid S)] \\
&\geq \text{Var}[\text{E}(T \mid S)]
\end{align*}
so $\text{E}(T \mid S)$ is at least as good of an estimator as $T$ since they both have the same bias.  The importance of sufficiency here is that it guarantees $\text{E}(T \mid S)$ doesn't depend on $\theta$ and hence is an actual estimator.
I also wouldn't put too much emphasis on consistency.  Consistency is actually a very weak property and it doesn't mean much without at least knowing something about rates of convergence.  It does us no good for instance to have a consistent estimator that requires an effectively infinite sample size before it can be expected to be close to the real value.
A: 
Although many times we will get the same answer whether we use the MLE
  or MOM methods, sometimes it can be different, particularly for more
  complex probability models. In such cases, the MLE-based estimator is
  often considered the better one. The tradeoff is that finding the
  maximum likelihood estimator can be more difficult, and sometimes even
  require computational effort.

(Probability and Statistics for Bioinformatics and Genetics, Course Notes, Paul Maiste, The Johns Hopkins University)
