As far as the meaning goes, they are exactly the same.
The difference is in the register, and you can understand it from the etymology of each word. "Keep" is common speech, and it has a more direct, personal effect. You probably recognise "continue" as there's a similar word in Spanish. To the English mind, this sounds a little more formal and academic, so your example of "Continue bothering me and I will..." is understandable, but a bit "off". You need more direct language for a threat.