What it is
The problem of evil asks whether the suffering in the world counts as an argument against God: if God is all-powerful, all-knowing and wholly good, why is there evil?
A theodicy answers by giving the reason: for each evil, it describes what would justify God in allowing it. A defense claims less. It only tries to show that God and evil aren’t logically contradictory, without claiming to know God’s actual reasons.
How it shows up for me
This question is a big part of why I’m an atheist. No theodicy I read could explain a suffering child in a way I could say out loud to that child’s parent. I still respect the people who keep wrestling with it.