Requirement and Specification are standard English words. They don't mean anything different from the dictionary definitions.
A requirement is "something you need, or something you must have." A specification is "how to do something" or "how to make something."
TL;DR: (if I really had to) I would use "design specification" as a superset of "requirements"
My experience is that:
Requirement: tends to be used more commonly in the design phase of a product, and also imply a "absolute minimum".
E.g.: When people are on a drawing board and try to decide what the requirement for a new ...