I am interested in this question from my operator theory study, but I am terrible at seeing the difference between nets and sequences... Can anyone share an idea? Especially I prefer to visualize things.
"How would $K(\mathcal{H})$ (the compact operators) change if we took it to be the set of limits of nets of finite-rank operators (rather than the norm closure of $FR(\mathcal{H})$)?"
More generally, not necessarily having anything to do with operators or topological vector spaces, one can show (as a reasonable exercise) that in a complete metric space, any Cauchy net has a subnet which is a Cauchy sequence. Even if we only require that "completeness" means "sequential completeness", that Cauchy sequence will have a limit. And, then, one should show that the limit of that sequence is the limit of the original net.
If/when we take weaker topologies on various spaces of operators (e.g., "strong operator topology" on continuous ops on a Hilbert space, ... which is strictly weaker than the operator-norm topology), some of these are not complete-normable, so subtler notions of completeness are needed. In fact, the strongest... and plausible-sounding... notion, that every Cauchy net converges, already fails to hold in the weak dual topology on the dual of a separable (but not finite-dimensional) Hilbert space. (No, the proof I know is not very direct or constructive.)
(Apart from other choices, such as talking about bornologies or other ways to look at nearness structures on vector spaces of (generalized) functions, it appears that the notion of quasi-completeness, or local completeness, both holds for most natural function spaces, and is sufficient for vector-valued integrals and many other things.)