I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and maybe this doesnt't help my intuition), but I am having trouble seeing where in the setup they differ.

To my understanding, the singular chain complex on a space $X$ consists of the free abelian groups generated by the sets of $n$-simplices in X, where an $n$-simplex in this context is a continuous map $\sigma : \Delta^n \to X$ from the standard geometric $n$-simplex $\Delta^n$ to $X$, with boundary map $\partial_n = \sum_{i=0}^n (-1)^i d_i$ where $d_i: C_n (X) \to C_{n-1} (X)$ is the $i$th face map ("deleting" the $i$th vertex). The singular homology groups are then the homology groups of this complex (ie. $H_n(X)=\ker(\partial_n)/\text{im}(\partial_{n+1})$).

Now for the simplicial homology, we have a simplicial complex $S$, which is a set of (abstract?) ordered simplices, such that a face of any simplex in $S$ is itself a simplex in $S$. Then we form the simplicial chain complex where $C_n(S) = \mathbb{Z}[S_n]$, where $S_n \subset S$ is the set of $n$-simplices in $S$, i.e. the free abelian group generated by $S_n$. This complex has boundary operator $\partial_n = \sum_{i=0}^n (-1)^i d_i$, where $d_i$ is the $i$th face map. The homology groups of this is $H^\Delta_n(S) = \ker(\partial_n) / \text{im}(\partial_{n+1})$. Now for this to make any sense in a topological framework, we have the realization of $S$, $|S| = \coprod (S_n \times \Delta^n) / (d_i \sigma, y) \sim (\sigma, d^iy)$ for all $(\sigma, y) \in S_n \times \Delta^{n-1}$ and $d^i$ is the coface map. (As I understand it, $S$ is a blueprint of how to "assemble" the geometric $n$-simplices to form a space). And then of course, if you want to talk about a specific space $X$, you need to find a simplicial complex $S$, whose realization is homeomorphic to $X$.

I can see very well that these two are two very different ways of building up the framework, but what I don't understand is where in practice it differs. Don't they both require that you find a way to divide $X$ into $n$-simplices? The only difference I see, is whether you map from $\Delta^n$ into $X$ before or after you form your homology groups, but there must be something I'm missing...