A great deal......A destroyer is much smaller and less armed than a Heavy cruiser.
A heavy cruiser ( or battle cruiser, depending on the Nation ) is a Smaller version of a Battleship...which was considered the largest combat ship of the time ( Excluding length....Aircraft carriers were longer) and carried the largest guns.
Destroyers on the other hand, are significantly smaller, faster, and have much lighter armor and weapons.... their primary function was to serve as escort vessels for merchant chips, battle groups, and as sub chasers.
They generally ran about 2000-2500 tons, and about 350-360 feet long, carried torpedoes, and depth charges in addition to their smaller deck guns.
Behind the destroyers in size were frigates ad Corvettes.
on the left....destroyer....center, battleship, on the right, heavy cruiser
https://s-media-cache-ak0.pinimg.com/736x/25/5b/88/255b88504db4aa130742f8ffc9750bc2--heavy-cruiser-korean-war.jpg