Of course the parameter estimates will be different from ML and MM. For one thing, the ML estimate is some sort of robust mean, with tails downweighted, as opposed to MM-based brute force arithmetic mean.
First of all, for the MM to work, you will need to have higher order moments to ensure that the sums necessary for the MM converge. E.g., for the sum of the fourth moments to be asymptotically normal, you need to have finite 8th order moments; for it to converge in the sense of the law of large numbers, you may need just another bit of a moment on top of the fourth, but at any rate, you have limitations on the degrees of freedom of your $t$ distribution. If you come to the data without knowing that your d.f. is >10, then you can find yourself in a situation where the MM estimator is not even consistent. High variability in the estimates, even in the situation when the estimator is defined and consistent, will also make MM less asymptotically efficient than ML.
ML does not have these theoretical problems, but in practice, it is found through numerical maximization, and in the particular case of $t$-distributions is prone to multiple local maxima. (By particular case, I mean that generally people speculate whether you have multiple local maxima or not, but in the case of $t$-distribution, existence of local maxima can be demonstrated analytically.) So you have to try a bunch of different starting values for it. Having a strong functional dependence between the variance and the shape parameter slows down convergence, as well.
As a bottom line, I don't see much use for MM... except probably for starting points for ML.