Let me explain it like this:
Creativity is the generation of new ideas. Either new ways of looking at existing problems, or the discovery of new opportunities.
Innovation is the exploitation of new ideas. It is the process that carries a concept through to new products, services, or ways of operating the business.
Design is what links creativity and innovation. It shapes ideas so that they become practical and attractive propositions for users or customers.
Design is the act of translating an idea into a blueprint, while keeping the solution focused on the customer.
I'd love to give you a reference for ‘creativity', ‘innovation' and ‘design', as these definitions are not my own. Unfortunately however, these are notes I picked up in a graduate course on design and innovation management, and I have no idea where they originated.