According to Wikipedia a modular form is a holomorphic function $f:\mathbb H \to \mathbb C$ satisfying quasi-invariance under the action of $SL(2, \mathbb Z)$ on $\mathbb H$ and a growth condition.
Also, according to Wikipedia an automorphic form is a "nice" function on a group $G$ to $\mathbb C$ satisfying invariance under a discrete subgroup $\Gamma \subset G$ and a growth condition.
I don't see how a modular form is an automorphic form. $SL(2, \mathbb Z)$ is not a subgroup of $\mathbb H$, right?
Edit: I don't even see how $\mathbb H$ is a group. It is not a group under normal $+$ or $\times$.
The functions $f:\mathbb H\to\mathbb C$ and $g:G\to\mathbb C$ in the two definitions do not correspond directly, because $\mathbb H$ is not a group.
But dig into the modular forms Wikipedia page and there is an equivalent definition, Definition in terms of lattices.
A lattice on the complex plane can be defined by a pair of complex numbers $u=u_1+u_2 i, v=v_1+v_2i$ with $u_1v_2-v_1u_2> 0.$ This can be seen as a matrix $$A=\begin{pmatrix}u_1&v_1\\u_2&v_2\end{pmatrix}\in GL(2,\mathbb R),\det A>0$$ But if $B\in SL_2(2,\mathbb Z)$ we get that $BA$ defines the same lattice as $A$ does, and $\det BA=\det A>0.$
This means we can take $G=\{A\in GL(2,\mathbb R)\mid \det A>0\}$ and $\Gamma=SL_2(2,\mathbb Z)$, and a function $F$ from the set of lattices to $\mathbb C$ can be seen as a function $F':G\to\mathbb C$ with the property that $F'(\gamma g)=F'(g)$ for all $\gamma\in \Gamma, g\in G.$
So, a modular form can be seen as a map $F'$ of a group $G$ to $\mathbb C$ where the distcrete sub group $\Gamma\subset G$ acts invariantly.
Now, the only question is if this $F'$ can be seen to satisfy the requirements of an automorphic form, based on the modular form requirement from which it arises.
I won't go into those details, but suffice to say, the reason studying modular forms is a case of studying automorphic forms in not based on a simple read of the the more elementary definitions of the two.