Quick answer
Dethatching removes the spongy layer of dead stems that builds up on top of the soil. Aeration pulls cores out of the soil to relieve compaction. Thatch is a surface problem; compaction is a soil problem. Some lawns need both.
How to Tell Which You Need
Cut a small wedge of turf and look at the brown layer between the green blades and the soil. If it's thicker than about half an inch, thatch is a problem.
If the soil itself is hard and water runs off, the issue is compaction, and aeration helps more.
Timing for Warm-Season Lawns
Do both while Bermuda and zoysia are actively growing, typically late spring to early summer, so the lawn recovers quickly.
Doing Both
When both are needed, dethatch first, then aerate. That clears the surface so cores pull cleanly and water and nutrients reach the soil.