Your intuition is exactly correct - it's not just 'tone' it's 'deeper than that'.
Codex is a 'poor communicator' - which matters surprisingly a lot in these things. It's overly verbose, it often misses the point - but - it is slightly stronger in some areas.
Also - Codex now has 'Spark' which is on Cerebras, it's wildly fast - and this absolutely changes 'workflow' fundamentally.
With 'wait-thinking' - you an have 3-5 AIs going, because it takes time to process but with Cerebras-backed models ... maybe 1 or 2.
Basically - you're the 'slowpoke' doing the thinking now. The 'human is the limiting factor'. It's a weird feeling!
Codex has a more adept 'rollover' on it's context window it sort of magically does context - this is hard to compare to Claude because you don't see the rollover points as well. With Claude, it's problematic ... and helpful to 'reset' some things after a compact, but with Codex ... you just keep surfing and 'forget about the rollover'.
This is all very qualitative, you just have to try it. Spark is only on the Pro ($200/mo) version, but it's worth it for any professional use. Just try it.
In my workflow - Claude Code is my 'primary worker' - I keep Codex for secondary tasks, second opinions - it's excellent for 'absorbing a whole project fast and trying to resolve an issue'.
Finally - there is a 'secret' way to use Gemini. You can use gemeni cli, and then in 'models/' there is a way to pick custom models. In order to make Gem3 Pr avail, there is some other thing you have to switch (just ask the AI), and then you can get at Gem3 Pro.
You will very quickly find what the poster here is talking about: it's a great model, but it's a 'Wild Stallion' on the harness. It's worth trying though. Also note it's much faster than Claude as well.