light: light cruisers are all about one thing: speed. they are the thoroughbreds of large military vessels. their armament and armor are limited and/or undersized to achieve high speeds. you know how corvettes were prior to patch 1.3? strong in numbers but helpless on their own for anything other than running away. CLs are like that- but much larger and able to take some hits. in stellaris, a CL would be a very flexible ship, best utilized as a scout, space cleanup ship (getting rid of basic threats single handedly), or suicidal distraction. it's armament is weaker (best for reusing older, less power intensive designs), it's base speed is much faster, base health much lower, and cost far lower. the Constellation-class from star trek is a prime example.
battle: battlecruisers vary slightly between sci-fi franchises but they generally follow the ship's historical use: armament roughly on par with a battleship, ridiculously fast, poorly armored. battlecruisers are the counter to heavy cruisers (they were nicknamed cruiser killers for a reason you know). compared to the CAs, they have less health, greater speed, and a ton of armament for a slightly higher cost. and considering that BBs in stellaris are now essentialy mobile artillery, the BC would fit well as essentially the machine gunner role complementing the tank and artillery in a late-middle/late game brawl with FEs and crises.