There is no difference in quality between a company's products marketed under its own brands and those that are brand-engineered for other retailers. Sometimes there are minor differences, such as the functionality of a feature, or the availability of a particular feature. For example, Whirlpool might have two switches with two options each that give you four options total (2X2), while Kenmore might bundle the same functionality into one switch giving you four options total.
One important note, though, is that part prices can differ between pure-branded and brand-engineered products. Let me give you an example. If your Frigidaire front load washer that appears as Kenmore at Sears, and GE at Home Depot needs repairs, you will pay a higher price for the part if it has a GE part number as compared to it having a Frigidaire part number. It is the same part, but the part number that appears on the box depends on the channel of distribution that it takes on the way to you. Most people don't know this, and are unable to do the math before making a purchase, but I would personally stay away from products made by other companies that are brand-engineered to the GE nameplate.
When it comes down to it, nobody will ever know how clean your clothes really are, just by looking at your or your machine, and you probably won't be able to tell either. Most people buy way too many bells and whistles, when all you probably need is perhaps 2 speeds and 2 water temp options. Otherwise, most machines do pretty much the same thing.