In my experience (very subjective, obviously) for backend/"logical" tasks Codex seems to outperform Claude.
For front-end/UX related tasks Claude wins easily.
Overall, Claude does seem to be a little better in other areas too.
Codex's biggest advantage in my personal opinion, however, is usage. I think maybe once in several months did I even get close to hitting my limit with the $20 plan.
With Claude, however, I feel like I can sneeze and half my weekly usage is gone. Same $20 price tag.
That's been my experience, I'm sure it differs user to user though.
The underlying motivation is to avoid Big Ai as much as possible. Incentives are not aligned.