Both the theory of moduli of curves and Teichmüller theory seem to be concerned with the moduli of Riemann surfaces.
However, they appear to belong to different fields within mathematics.
Could someone explain in which ways these approaches to studying Riemann surfaces differ and wherein their similarities lie?
Also, I have seen that in both theories compactifications appear to play an important role and I would be interested to hear if these are somehow related. I have heard that to compactify, one has to add curves of a different genus. Does this apply to both theories?
The subjects you are alluding to belong to the same field of math, it is just that analysts and algebraic geometers have different ways to deal with the moduli spaces and their compactifications. Analysts like to think of the moduli space as the orbifold quotient of the Teichmüller space by the mapping class group, $T_{g,n}/Mod_{g,n}$, while algebraic geometers like to think of the moduli space ${\mathcal M}_{g,n}$ as a certain Deligne-Mumford stack. The latter viewpoint proves that the coarse moduli space is an algebraic variety; an analytical proof of this fact was given by S.Wolpert who analytically constructed an ample line bundle on the coarse moduli space.
It is a bit hard to pin down a canonical reference to the isomorphism of the orbifold $T_{g,n}/Mod_{g,n}$ and the stack ${\mathcal M}_{g,n}$. Sadly, it does not appear in any textbook on Teichmüller theory. Hopefully, it will, one day...
In the case $n=0$ (compact Riemann surfaces), a proof of this isomorphism is due to Grothendieck:
A. Grothendieck, Techniques de construction en géométrie analytique. Séminaire H. Cartan, 13ème année: 1960/61, Exp. 7 (1962) pp. 9–17.
The general case ($n\ge 0$) was proven independently in
C. Earle, On holomorphic families of pointed Riemann Surfaces, Bulletin of the American Mathematical Society, 79, 1973, pp. 163–166
and
M. Engber, Teichmüller spaces and representability of functors. Trans. Am. Math. Soc. 201 (1975) pp. 213–226.
A detailed account of Earle's argument (with a proof of a more general result, dealing with Riemann surfaces of infinite type) appeared in
C. Earle, R. Fowler, Holomorphic families of open Riemann surfaces.
Mathematische Annalen, 270 (1985), pp. 249–273.
Accordingly, there are also two viewpoints on the compactification of the moduli space: Analytical (going back to a work of Bers as well as an unpublished work of Earle and Marden) and algebro-geometric (due to Deligne and Mumford). These are related in the paper
J. Hubbard, S. Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves. J. Differential Geom. 98 (2014), no. 2, pp. 261–313.