It goes like this:

Codex is an outsourcing company, you give specs, they give you results. No communication in between. It's very good at larger analysis tasks (code coverage, health etc). Whatever it does, it does it sloooowwwllyyy.

Claude is like a pair programmer, you can follow what it's doing, interrupt and redirect it if it starts going off track. It's very much geared towards "get it done" rather than maximum code quality.