Storytelling was originally an advertising technique but is now associated with brand strategy. It is also called, for added polish, Strategic Storytelling. This is a concept and a professional practice, as opposed to a copywriting technique. As such, it is a form of Strategic Consulting.
Strategic storytelling calls for creating stories about how people experience a brand. Thus, things like features, benefits, and UVP are revealed through characters’ experience with them in a brief story rather than through, for example, bullet points. Story characters could be customers, constituents, employees, owners, execs, etc.
Other common business-world synonyms include the slightly more abstract concepts, Narrative Messaging and Strategic Narrative.
The primary difference between storytelling in advertising and branding is timeframe - in advertising, stories are meant to effect buy (or buy-in) right away; in branding, the buy can happen at any time in the future.
Good storytelling is more concerned with humanizing a brand and entertaining its readers, rather than forcing an understanding of value proposition.
Strategic (or brand, advertising, political, business, etc.) storytelling is not to be conflated with actual storytelling, which is theoretically free of business or political motives and thus free to explore whatever it wants, such as psychological truth. But this conflation happens often; thus, it is a controversial-at-worst, awkward-at-best term. Ogilvy did not conflate the two; he talked about, "storytelling in advertising". He also offered maybe the best-ever example of strategic (political, in this case) storytelling:
"When Aeschines spoke, they said, 'How well he speaks.' But when Demosthenes spoke, they said, 'Let us march against Philip'"