Keep on: "to continue doing something, or to do something many times."
Go on: "a) to continue doing something or being in a situation. b) to continue without stopping".
From Longman Dictionary of Contemporary English
Per my comment to the question, in most contexts, all three of keep on / go on / continue [present participle] are synonymous and interchangeable (arguably I've given them in descending order of "informality").
But in this context I would very much prefer keep on (and definitely wouldn't use continue)...
Maybe it's just a personal thing, but I feel to keep on has more overtones of be persistent, dogged, whereas being told to go on often means little more than "don't stop".
There are probably other contexts where one term is commonly understood to be more (or less) suitable than the others, but nothing else comes to mind at the moment. So comments welcome.
I agree with FumbleFingers that 'keep on' implies a measure of dogged persistence, where 'go on' could describe cruise control. Neither strikes me as more or less formal than the other, and both are good English. Slight difference in flavor and/or implication, depending on how fine you want or need to cut it.