What exactly is the difference between "Keep on doing" and "Keep doing"?
I would say that "keep on doing" is ever so slightly more emphatic than "keep doing".
Nothing except the extra word in the middle. By stretching my imagination, I can see 'keep doing' as slightly more formal, since phrasals tend to be colloquial.
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Really not? Emotion: surprise
Really not. Other opinions?
 taiwandave's reply was promoted to an answer.
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There just may be a slight tendency for one form (with "on") to stress continuity of action while the other form (without "on") is used slightly more to stress repetition.

I asked her a question. She never even looked up. She just kept on reading as if I weren't there.
Every morning she went to the library and checked out more books. She just kept reading more and more books all summer long.

Peter kept on eating while we talked.
It was very frustrating that week. We set the trap every night, but the mouse kept eating the cheese without springing the trap.

The problem with this theory is that it would be nearly impossible, in my opinion, to find a case where one form could absolutely not substitute for the other. Still, if the examples above are representative (and I have no proof that they are), it is much more natural to add the "on" in the 'repetitive' examples than to remove it in the 'continuous' examples.

Maybe someone can investigate whether with an intransitive verb, the "on" is felt to be more necessary than with transitives. "kept sleeping?" "kept thinking?" "kept worrying?"

Calif Jim