This question already has multiple nice answers, but I am going to add one more thing which isn't quite covered by the existing posts.

One distinctive advantage of the $\Gamma_0(N)$ and $\Gamma_1(N)$ subgroups comes from *new-vector theory*: for these levels, we can split up the space of modular forms into an "old part" (built up in a rather explicit fashion from $\Gamma_1(M)$ with $M < N$) and a "new part" (the stuff that genuinely belongs at level $N$). Then, for newforms, we have very strong uniqueness results like the strong multiplicity one theorem (which says that if two Hecke eigenforms in the new subspace have the same Galois representation, they are equal). 

These properties of $\Gamma_1(N)$ would break down if you substituted some other random family of congruence subgroups in place of $\Gamma_1(N)$. Moreover, finding families of subgroups that give you similar multiplicity-one results for bigger reductive groups in place of $GL_2$ is a highly nontrivial question and an active topic of research right now.