"Go on" and "keep on" can both mean "continue". "Go on" occasionally just means "go", as in alawton's example.
In a long expression like "I need to keep on studying" the usage is the same. But the shortest expression using "go on" is "Go on.", whereas the shortest expression using "keep on" is "Keep on it." These two phrases are pretty similiar as well, but there are some cases where you would only use one or the other.
You wouldn't say "Keep on." You could say "Go on it." but in that case you wouldn't be using the expression "go on", you'd just be using the verb "go" with the phrase "on it", so the meaning would be different.