The band refers to how many bands (blonde/black) are visible in the brush knot. If you look at a two band brush, the base of the knot is black and the top of the knot is blonde. You will see a blonde, black and more blonde in a three band brush. There are noticeable performance differences between the two type of knots as well, as you noted when using the two band brush. Two band is not better than three band, but they both have their advantages. I am on my ipad otherwise I would post a side by side picture to show you what I am talking about. Hopefully someone will chime in and post a few pictures.
Agreed with Cessnabird - I would only add that as a general rule - 2 band tend to be a bit more "stiff" as opposed to 3 band that tends to be a bit more "floppy"...stiff and floppy probably aren't the best terms to use, but the 2 bands are generally better for hard soaps and the 3 bands tend to shine a bit more for bowl lathering. There is some "cross-over" for sure, those are just general descriptions.