I'll discuss things which are more applications of the mapping class group to moduli space rather than Teichmuller theory per se, but of course this is all tightly connected.

One of the big applications of this point of view is to the cohomology of moduli space. The moduli space of curves is not quite a classifying space for the mapping class group because the action of the mapping class group on Teichmuller space is not free, but the problem all comes from finite order elements. One can think of moduli space as a "rational classifying space" or an "orbifold classifying space" for the mapping class group. The upshot is that the group cohomology of the mapping class group is identical to the cohomology of moduli space with $\mathbb{Q}$ coefficients.

I will try to give a brief survey of this field, but it is huge and I will omit a lot of important work.

There is now a lot known about the group cohomology of the mapping class group. The most spectacular is the resolution by Madsen-Weiss of the Mumford conjecture giving the rational cohomology ring in a stable range. This is certainly not known via algebro-geometric methods.

This was proceeded by many older results. The most germane come from a series of papers by Harer in the '80's which (among other things) do the following:

1) Show that the cohomology stabilizes as the genus increases.

2) Calculate the Euler characteristic. This really is not a theorem about the mapping class group, as the proof uses a certain triangulation of moduli space rather than group theory. However, this triangulation definitely comes from Teichmuller theory rather than algebraic geometry, and it is still part of this same circle of ideas.

3) Make a number of low-dimensional calculations (up to degree 3 in published work and 4 in unpublished work).

The calculation of $H_2$ by Harer in particular is the key to calculating the Picard group of moduli space.

These low-dimensional cohomology calculations can now be (basically) done via algebraic geometry. See the paper "Calculating cohomology groups of moduli spaces of curves via algebraic geometry" by Arbarello and Cornalba. Thus the Picard group of moduli space can now be calculated via algebraic geometry.

A more recent application of this point of view comes from work of myself which calculates the Picard groups of the moduli spaces of curves with level structures (see my paper of the same title). I think it would be very interesting to try to make this same calculation using algebro-geometric methods, but I have no idea how to do so.