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.