Sets distinguish between containment and inclusion.
A⊆B means A is included in B (subset).
A∈B means A is contained in B (member of or element of)
I've located this graphic depiction of topological containment:
However, looking at the points as sets, it isn't clear to me how this differs from inclusion.
The word "contains" in this case is not being used to suggest set membership. If we view graphs $G$, $X$, and $Y$ as topological spaces, then $X$ and $Y$ are homeomorphic topological spaces, and $Y$ is a subspace of $G$: a subset of $G$ with inherited topological structure.
From the combinatorial point of view, it doesn't make much sense to think of elements or subsets: these are defined for sets, not graphs. We have the notion of a subgraph (which is similar to a subset: a subgraph of $G$ means we choose a subset of the vertices of $G$, and a subset of the edges of $G$ between them) and here, we relax that notion slightly: instead of asking for $G$ to have a subgraph isomorphic to $X$, we only ask that $G$ has a subgraph isomorphic to a subdivision of $X$.