Why do we use nets instead of sequences in a topological space that is not first countable?

In topology, we can't characterize things like closure and continuity using sequences when a space is not first countable. What is it about nets that allows us to do these characterizations, replacing "sequence" with "net"? Perhaps it has something to do with the net being defined on a directed set instead of the naturals, which then begs the question of why directed sets fix things. Further more, what is it about not being first countable that ruins our ability to use sequences?