Cayenne Pepper is made from the dried pods of pungent chili peppers so in my mind this would be different than red pepper.
Red Pepper is the dried, ripened fruit pod of Capsicum frutescens, one of the most pungent Capsicums. It is sometimes referred to as Cayenne Red Pepper, having been named after the high heat chilies grown in the vicinity of the Cayenne River in French Guiana.
While it's referred to as cayenne red pepper, I do believe they are different.