I've had the chance to learn about nets, though every statement I was exposed to didn't seem to be useful in practice.
For example, the fact $x \in \bar{A}$ iff there exists some net $(x_\alpha)_{\alpha \in J}$ such that $x_\alpha$ converges to $x$, doesn't seem to be useful in practice because if one can construct such a net, than one can also prove directly that every neighboorhood of $x$ has a non empty intersection with $A$.
Question: Are examples of results about concrete spaces (or otherwise — spaces that have some specific property that is not described by the concept of nets) being obtained using nets, and that could not be obtained using sequences, where nets actually prove to be an efficient tool?
First comment: anything you can prove using net convergence can be proved in other ways; for example, using filters (see, e.g., Kelley, General Topology, p.83). In principle, you can always go back to arguing directly in terms of open sets. After all, net convergence (or filter convergence) is defined in terms of open sets, so you can always "expand out" the references to nets.
But this can be cumbersome. Nets are useful for spaces whose topology is not first countable, so sequential convergence and net convergence don't mean the same.
One example: the weak* topology on the dual of a Banach space. By the Banach-Alaoglu theorem, any infinite sequence $\{f_n\}$ on the unit ball has an accumulation point. But there might not be a convergent subsequence, $f_{n_k}\rightarrow f$. However, there always will be a convergent subnet.
Another example: let $H$ be a Hilbert space, and let $L(H)$ be the space of bounded linear operators on $H$. $L(H)$ has several important topologies; one is the weak operator topology. Let $A\subseteq L(H)$, and let $A'$ be the set of all operators that commute with all elements of $A$. Then $A'$ is closed in the weak operator topology. Proof: let $S\in A$ and let $T_\alpha\rightarrow T$ be a net with all $T_\alpha\in A'$. Then $$ ST = \lim ST_\alpha = \lim T_\alpha S = TS $$ so $T$ commutes with $S$ too.
Of course, some things need to be proved for the this computation to work (for one thing, multiplication of operators is weakly continuous in each variable separately).
But the point is, once you've developed the "infrastructure", proofs using nets look just like proofs using sequences. That makes them very appealing and intuitive (for many people, including me).