Identity designers set the direction of the visual elements that make up your brand; your name, logo, color palette, typography, tone of voice, online and offline appearance. These are unique to your business and define your brand, as well as it's success.
Branding is the sum of all efforts you make to interact with your customer and is a part of your marketing strategy. Choosing a logo for example, is one of these efforts. A logo is at the core of a brand, and should be strategically designed and chosen from a marketing perspective.