There is a lot of difference between a destroyer and frigate and both serve their roles equally well in any Naval Force.
Frigates are generally medium-sized ships ( in comparison to Destroyers ) and are specialized in a particular role like Guided Missile Frigates ( where they will be carrying cruise missiles or Surface-to-Air Missiles ) and Anti-Submarine Frigates ( where they will be equipped with improved sonars to detect enemy submarines and weapons such as Depth Charges , Torpedoes and Anti-Submarine Rockets ) . Frigates lack the multi-mission capability like Destroyers and usually comes in Protection of Shipping (POS) roles.
Shivalik Class Frigate
Modern day frigates weigh almost 2000-5000 tonnes and are equipped with a variety of weapons like SAM , Cruise Missiles and Anti-Submarine Warfare assets.
Destroyers on the other hand are heavy surface combatants and are heavy at 5000-10000 tonnes and equally expensive. These pack the most offensive firepower for any Navy packed with Cruise Missiles , Anti-Ship Missiles , SAMs , ASW assets along with top-notch sensors and radars and thus literally pack the punch of any naval force on a single ship.
Kolkata Class Destroyer
If we compare their fighting capabilities , then Destroyers easily edge out Frigates thanks to its better armaments and attack capabilities while frigates mostly acts as a Fleet protector and as a “Flagship” or “Convoy” in modern sense. Frigates are also cheaper to manufacture and operate than Destroyers.
Both Frigates and Destroyers are needed for a Blue-Water Navy as they have capability to operate in high threat environment and thus are well coordinated in such roles.
NOTE : Many Navies now building “Frigates” which in practical terms are as sized and capable as Destroyers like German F-124 and Dutch De Zeven Provincien class.
F124 Sachsen-class Frigate
So at end all these are text-book terms used for classification purposes rather than actual capability terms.