For some specific problems, it saves you a lot of time. For instance, one of the properties of the moment generating function is that $M_{X+Y}(t)=M_x(t)M_Y(t)$ if $X$ and $Y$ are independent. So to figure out the distribution of $X$ and $Y$ you could do it through a convolution of their densities, which is usually long and tedious, or through moment generating functions.

Also, to compute the $k$-th moment of $X$ you can just do $M^{(k)}(0)$, which also can save you time if you need to compute a lot of moments.

For some specific problems, it saves you a lot of time. For instance, one of the properties of the moment generating function is that $M_{X+Y}(t)=M_X(t)M_Y(t)$ if $X$ and $Y$ are independent. So to figure out the distribution of $X$ and $Y$ you could do it through a convolution of their densities, which is usually long and tedious, or through moment generating functions.

Also, to compute the $k$-th moment of $X$ you can just do $M^{(k)}(0)$, which also can save you time if you need to compute a lot of moments.