How did DD and DE destroyers differ?

Answer 1

DD is the code for an actual "Destroyer" these are small, fast ships were originally designed to work with a battleship fleet screening it from attacks by Torpedo Boats - they were originally called "Torpedo Boat Destroyers" which became shortened with time. Over time the role changed as well since aircraft and submarines became more of a threat than torpedo boats but the basic use was the same - protect the large (expensive!) battleships, and later aircraft carriers, from attacks by small ships/boats/aircraft.

These ships gradually become larger and more powerful in their own right and in modern navies are often used as, and have sometimes replaced, "Cruisers" - ships operating independently from the fleet as a raiding/scouting force.

DE is a US navy code for a "Destroyer Escort", other navies used different names - the Royal Navy called them "Frigates" the French called them "Corvettes" but basically they are smaller, slower, cheaper than "real" destroyers. These ships are not designed to work with a battle fleet so don't need the same speed and endurance, they were to work as convoy escorts or submarine hunters - they often look quite similar but tend to be smaller and armed more for sub hunting.