asciinema actually records output, this relies on the environment being exactly as it was the first time you ran it. If one command fails or has different output, other things might not work.
This also cannot follow cli sessions. If you open, say, a REPL, you're on your own.
That said, I would love for asciinema to get some "type to progress" capabilities.
First difference seems to be that do it live allows manual control of progressing through the session where as asciinema doesn't.
What else?