In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times A$; however in, for example, the fundamental theorem of finitely generated abelian groups, we normally write that every such group is isomorphic to one of the form
$$
\mathbb{Z}^n \oplus \mathbb{Z}_{r_1} \oplus \cdots \oplus \mathbb{Z}_{r_t} 
$$
where $\mathbb{Z}^n$ now means $\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}$.

From an intuition perspective, and in the sense of sets, is this more or less the same as $\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z}$? (Bear in mind I am normally using these ideas in relation to homology groups.)

As long as you restrict to finite index sets, the direct sum and the direct product of commutative groups are identical.

For a general index set $I$, the direct product of commutative groups $\{G_i\}$ is the full Cartesian product $\prod_{i \in I} G_i$, whereas the direct sum $\bigoplus_{i \in I} G_i$ 
is the subgroup of the direct product consisting of all tuples $\{g_i\}$ with $g_i = 0$ 
except for finitely many $i \in I$.

The coincidence of the direct sum and the direct product in additive categories can also be explained in categorical terms, but I'll let someone else take a crack at that if they so desire.  
