I'm not sure what you think the distinction between "permit" and "allow" is.
Permit | Define Permit at Dictionary.com
The first definition of permit is "to allow."
What does "sign off on" mean in the following? Permit or allow?
It turned out to be an exclusive Q&A with the Apple CEO. One Jobs didn't necessarily sign off on (and would never after all of this). Ethically, the whole thing just seems flaky.
I thought those two words are almost the same.
So you agree "sign off on" means "to allow" in the context, don't you?
Since this was something involving him, I would say "agree to" is a better definition. He wasn't just going to allow something, he was going to have to do it.
But in some companies there's actually a piece of paper involved. If you sign off on something you both agree to it and sign the paper - reording the fact that you've agreed. I don't know whether in Apple at the time this sort of 'sign-off' was involved.