Marketing consists of all activities associated with creating a customer for your product. So in business marketing is everything, really: "Business exists to create a customer" said Peter Drucker. It's like a microphone in that it builds awareness.
Branding is a subset of marketing, a meta-technique applied to all the assorted activities done in marketing's name. It's what makes you unique - your fingerprint. Branding is "the way we do things," "who we are," "our personality."
Branding is the long-term action of creating perceived value: adding value to your product over and above commodity.
Branding activities have less short-term ROI but they function as long-term insurance when the product is getting best up by the market.
You don't abandon your friends easily.
Think of marketing as generating positive attention for your product and branding as building a cushion underneath it. An insurance policy.
That cushion is that people want to buy from someone they know and trust - and that is a sense you build through branding.
You do both at the same time: Marketing and branding activities overlap and integrate.
The influence of branding is to integrate disparate marketing activities into an overarching message. When activities are coordinated the message is more powerful.
Branding enhances marketing when the message is focused, differentiated, relevant, and consistent. And of course credible.
Marketing techniques typically include: advertising, PR, trade shows, social media, word of mouth, product placement and employee-focused or internal communication. It also includes research and development.
When you do marketing absent branding the message does not stick.
Branding activity: Create a personality around the mouthwash such that kids want to integrate into their daily lives and even pay double the price of generic.
Marketing activity: Get the mouthwash in front of kids in an appealing but safe way. Make it indispensable.
Ideally there is one person (a benevolent dictator) whose vision runs the whole show. That way the focus is truly singular.
* The above was originally posted as a response to a request to differentiate marketing and branding, posted on Quora.com.