A reaction in both directions may be equilibrating, but is not necessarily equilibrating (like if the system is being excited), and it certainly isn't necessarily at equilibrium.
For example, if one puts pure ammonia gas in a container, it will be far from equilibrium with respect to its decomposition products (nitrogen and hydrogen gas) but it will spontaneously proceed in the following direction:
$2\mathrm{NH}_3 \rightarrow \mathrm{N}_2 + 3\mathrm{H}_2$.
Some time after that, the rate of the reverse reaction becomes non-negligible;
$\mathrm{N}_2 + 3\mathrm{H}_2 \rightarrow 2\mathrm{NH}_3$,
so that both reactions simultaneously occurring can be summarized as:
$2\mathrm{NH}_3 \rightleftarrows \mathrm{N}_2 + 3\mathrm{H}_2$. That doesn't mean the system's at equilibrium, just that the forward and reverse rates are non-negligible.
Finally, the equilibrium condition (so that the concentration of all species remains constant over time) applies, and the forward and reverse rates become equivalent. That's the distinction that's specified with the double harpoon arrows:
$2\mathrm{NH}_3 \rightleftharpoons \mathrm{N}_2 + 3\mathrm{H}_2$.
I think the harpoon arrows are most often used to specify reaction rates, so the equilibrium condition, where the forward and reverse rates balance, is marked using them.