Both words can be used to describe when a person is frightened (although, typically, I initially think of shivering in the cold, and shuddering at a thought).
I also think of shivers lasting a little longer than shudders. I might say, "I was shivering at the bus stop for almost twenty minutes!" but, "When I stepped outside, I shuddered in the cold," referring to a momentary shudder.
Likewise, if a young child trembled in fright during a scary movie, I might say he "shivered through the movie," or "shuddered at a scene." However, I must point out there are no hard rules about this; these are just ways I typically use these two words.
Shiver is also used in the expression "Shiver my timbers!" – which I don't hear very often, unless someone is trying to do a humorous imitation of a pirate.