The direct sum ⊕\oplus versus the cartesian product ×\times

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

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.)

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 $i \in I$.