I use both.
Destroyers are cheap to build and can be specialized for Anti-sub/anti-Air/Anti-Escort work, but generally useful only for a single specialty. CLs, are also able to specialize, but the additional weapon/equipment slots gives them better versatility. I can effectively set up a CL to be primarily AS/AA/AE and still have enough space to give it a secondary role. The main guns provide better effect against other escort ships.
For example, a depth charge rack and sonar lets them hunt subs and a seaplane adds detection for both subs and surface vessels. The secondary purpose can be AA -- by adding more AA guns, AE -- add extra secondary guns (the same size as a destroyer's main weapons), and/or multiple torp tubes for use against capital ships. While they can be up to 3 or 4 times more expensive than a single destroyer, the added versatility and survivability is worth it to me.
Since the change to the ratio of escorts/capital ships to 3:1 (iirc); I use one CL and Three DDs as my standard escort unit.It's a useful mix in almost any type of ocean terrain. I can add a CA as well to make a small raider group or for heavy patrol or transport escort in dangerous water.