You already probably know the advantages of thinking of homology as being a functor of $X$. By allowing an abelian group $G$ as coefficients, it becomes a functor of two variables, the space $X$ and the group $G$. Now you can exploit the $G$-functoriality as well.
For example, the short exact sequence of groups
$$0 \to \mathbb Z \to \mathbb Z \to \mathbb Z/p\mathbb Z \to 0$$
(the second arrow is mult. by $p$) gives rise to the short exact
sequence of cohomology groups
$$0 \to H^i(X,\mathbb Z)/p H^i(X,\mathbb Z) \to H^i(X,\mathbb Z/p\mathbb Z) \to p\text{-torsion subgroup of } H^{i+1}(X,\mathbb Z)
\to 0.$$
(Here I have switched from homology to cohomology, since I find that latter a
little easier to think about, and easier to state these sorts of results about.)
So studying cohomology with coefficiens in $\mathbb Z/p\mathbb Z$ is closely
related to studying torsion in cohomology with $\mathbb Z$-coefficients. If $p$ is a prime (which is the case I was imagining, although the above is valid
for any integer $p$), then $\mathbb Z/p\mathbb Z$ is a field, and so we can investigate torsion in cohomology by studying cohomology with coefficients in a (finite) field. The latter investigation has technical advantages, e.g. Poincare duality if $X$ is a manifold.
Thinking of cohomology as a functor of $G$ as well as of $X$ leads to the idea
of sheaf cohomology (a sheaf of abelian groups is something like an abelian group that can vary from point to point over $X$). The possibility of taking $G$ to be a non-constant sheaf greatly increases the flexibility of cohomology as a tool.
Another example of utility of coefficients:
If $E \to B$ is a fibre bundle with fibre $F$, and $B$ is connected and simply connected, then
there is a spectral sequence (the Leray spectral sequence) $$
E_2^{i,j} := H^i(B,H^j(F,G) ) \implies H^{i+j}(E,G),$$
relating the cohomology of the base and fibre to that of the total space.
Note that even if we take $G$ to be $\mathbb Z$, the cohomology of the
base has coefficients in the cohomology groups $H^j(F,\mathbb Z)$, which
are likely not just equal to $\mathbb Z$.
If $B$ is not simply connected, then there is a similar spectral sequence,
but the coefficients are a locally constant, but typically non-constant,
sheaf on $B$.