One situation which highlights the difference between separability and non separability is that of an important class of locally convex spaces which fail many of the properties which are relevant to closed graph theorems (barrelled, bornological, etc.) but which do satisfy such a result for mappings into SEPARABLE spaces. This has interesting consequences, one of which we will now sketch.
A fascinating phenomenon involving non separability is the fact that many classical Banach spaces consist of bounded objects (continuous, uniformly continuous, measurable functions, linear operators) with the supremum norm and such spaces are ALWAYS non separable, or rather only separable under tight conditions. Thus separability of $C(K)$ implies that $K$ is metrisable, separability of $\ell^\infty$ requires finite dimensionality as does that of $L(H)$ and so on. There is a meta explanation of this fact which goes back to work of Saks, more precisely his original version of what is now known as the Vitali-Hahn-Saks theorem. Saks used a Banach-Steinaus-type theorem (rather than a closed graph theorem but these two themes are known to be closely related). The novelty of his approach lay in his use of the Baire category theorem (what else?), not on a Banach space but on the unit ball of an $L^\infty$ space topologised by the $L^1$ norm. The fact that the latter is not a linear space and the importance of translation in the Banach case made the proof more delicate and it required a geometrical condition relating the norms, now called $\Sigma$.
This led to the theory of Saks spaces—a unified approach to a class of Banach function or operator spaces for which the fact that the norm was too strong for certain applications (dual too large, bad density properties...) was remedied by replacing it with a weaker, but in these respects more suitable, l.c. topology (two-normed spaces, mixed topologies, Saks spaces ...). The paradigmatic example is, perhaps, the strict topology on the space of bounded, continuous functions on a locally compact space (R. C. Buck, et al.)—the elements of its dual space are precisely the bounded, tight measures.
The relevance of this to the question on the table is that one can use Saks’ ideas to obtain a closed graph theorem for linear mappings from certain Saks spaces into SEPARABLE Banach spaces. This means that in the concrete examples of interest, separability of the original Banach space forces a kind of collapse—the Saks space structure coincides with the Banach space one and this can only happen under restricted conditions.
Details are in the book ”Saks Spaces and Applications to Functional Analysis” which can be found online.