No disrespect intended, but your confusing branding and differentiation - both are marketing techniques, and may or may not be used in conjunction.
Your "Brand" is your recognition or evoked set in the consumer world. People will buy a Amsoil oil filter and pay a lot of money for it, because Amsoil is a well known brand of oil, even though they don't make oil filters. Currently Fram makes their filters.
Similarly Fram is also a well known brand. They have a differentiated product offering -- Tough Guard, ultra synthetic, Endurance, etc. Fram is the brand and the differentiation is in the product offering - good, better, best or whatever.
Taking these two together, the Amsoil oil filter and Fram Endurance are identical filters except the color and the label. One is much more money than the other. That is due to branding, not product differentiation. Again, would people pay an extra 8 bucks to get the same filter from Amsoil that they get from Walmart? Of course not, but they perceive the brand Amsoil provides a better product - hence they pay it - confused customer due to branding. This is 100% the purpose of Amsoil putting their name on it. Amsoil dealers could likely sell a Fram filter to their customers, but no one would pay the premium.
You can then mix the two. I can buy a Chevy, or a Buick. There both the same underneath. The Buick might have better appointments and materials. This is an example maybe of where the differentiation defines the brand. A buick is a Chevy with better seats - for example. Or maybe simply the perception of better seats. You see this a lot. A company will define a brand using differentiation. Then they will cheapen the brand over time, hoping people continue to buy it due to its previous reputation - ie again - confused customer due to branding.