Destroyers are generally used the for same purposes as a battleship, such as attacking naval units and buildings, especially transport ships and naval shipyards, though they are also used for their anti-air capabilities. Planes are one of the biggest threats to naval units, increasing the importance of having a couple destroyers in one's navy. They are also a cheaper option than battleships, and thus can be used when a player does not have a lot of money (or does not want to spend too much on their navy).
This is one of the 2 units that can damage submarines, the other being submarines themselves, making them useful for defending naval points when you haven't finished researching submarines.
After a couple minutes of comparing battleships and destroyers, we have discerned a key difference between the two: the shapes of the turrets on top. When observed closely, the Destroyer has one anti-air turret and one regular turret. This means that it will specialize against fights that include both air and navy. Battleships, on the other hand, are equipped with two regular turrets designed for fighting against ground/naval units. To summarize, bring destroyers to naval/air battles, and bring battleships to pure naval-only battles.(a better option of both worlds is to have a mix of destroyers and battleships)