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."
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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.