Every successful brand must have attributes or benefits that make them stand out in the hearts and minds of consumers and stand apart from their competition. These are the points of difference in the brand foundation and are fundamental to helping your brand win in the marketplace. They are what give your audiences a reason to turn your way versus the competition.
These can be rational, as in distinct product quality features or maybe even the fact that it's tied to a particularly strong parent brand/organization, or emotional, as in the way the brand makes its audiences feel.
Key here is that points of difference should be elements of the brand that competitors can't copy easily (think under six months). Unless the brand is willing to fight for trademarks, you can expect your assets to be copied.
One of my favorite examples of the nuances between points of parity and difference comes from the classic Miller Lite campaign. In this campaign they touted "Great Taste," a point of parity in the category plus "Less Filling," a point of difference at a time when consumers wanted fewer calories without sacrificing taste. Brilliant.