I can see why the difference may not be clear to you, as in Japanese, you use 非常 for both words. A contingency plan is simply an alternative, or a back-up plan in case something unexpected happens. The unexpected outcome may or may not be an emergency, and often in fact, is not - it is just a different outcome from the one originally planned for. Let's say you plan on flying home to visit your relatives in Golden Week, but you have a contingency plan (an alternative plan) to drive, or take the shinkansen, if the flights are cancelled due to bad weather, or you are unable to get bookings. 分かりましたか。