Let's consider the strong operator topology and the weak operator topology on bounded operators of a infinite-dimensional Hilbert space $H$. When they define these operator topologies, some authors use nets, some authors use sequences (assuming, explicitely or implicitely, that $H$ is separable, which means that one can nets and sequences indifferently).
What is the motivation of using nets instead of sequences to define these topologies? What do we gain in working with operator topologies on a non-separable Hilbert space?
Every topological space can be characterized in terms of its convergent nets. In an arbitrary topological space, a point $x$ is in the closure of a subset $A$ if and only if there is a net in $A$ converging to $x$.
First countable spaces, including metric spaces, have topologies that are determined by their convergent sequences. In an arbitrary first countable space, a point $x$ is in the closure of a subset $A$ if and only if there is a sequence in $A$ converging to $x$.
If $H$ is an infinite dimensional Hilbert space, regardless of whether or not it is separable in the norm topology, then the weak topology on $H$ is not first countable, and is not characterized by its convergent sequences alone. See On the weak closure for an example that illustrates this.
Similarly, the weak and strong operator topologies on $B(H)$ are not first countable unless $H$ is finite dimensional (in which case all of these topologies coincide with the norm topology). So the answer to the first question is independent of whether or not the Hilbert space is norm-separable, which may undermine your motivation for asking the second question, even if it doesn't answer it.