At least when I tried it last, Claude Cowork tried to spin up an entire virtual machine to sandbox itself properly - and not only is that sandboxing slow to start up, it also makes it difficult to actually interact freely across your filesystem. (Perhaps a feature, not a bug.)
Claude Code, on the other hand, has no such issues, if you've done some setup to allow all commands by default (perhaps then setting "ask" for rm, etc.).
Codex CLI is a TUI app, but Codex App is an actual desktop GUI app. If you actually look at the TFA, you'll see that all of the videos are of the desktop app.