What's different about two-band brushes is that the hair is stouter and stiffer than those in three-band types, and - perhaps most significantly - two-band hair is much, much rarer and harder to come by. That alone is going to up the "gotta have it" factor for a lot of people. Having said that, there are those, like myself, who genuinely prefer two-band hair, irrespective of its elitist appeal. Back when two-band brushes were relatively plentiful, and obtaining them was merely a matter of paying for them, I decided I genuinely preferred two-band brushes after trying them toe-to-toe with some outstanding three-band brushes. All things being equal (e.g., same loft, same knot thickness and density, same crown shape, etc.), a two-band brush will be stiffer and will not splay out as much as a three-band version of the same brush, thus holding its shape better and remaining more compact on the face. A two-band brush is also more resilient, and has a more "muscular" feel on the face. Some people would describe this quality is a "scrubby" or "scratchy" feel. I personally do not find my better two-band brushes to be scratchy. The trade off is that two-band brushes often don't feel as soft or luxurious as the best, softest three-band brushes. For me, the trade-off is a worthwhile one, as I don't like brushes that splay out and have a softer, mop-like feel. While I could live my life just fine using the better three-band brushes in my collection, I prefer my two-bands, and would seek to replace them ahead of my three-band brushes. Whether or not two-band brushes are better for use with different lathering agents (i.e., creams versus soaps) is subjective. Generally speaking, I find I can get equal quality lathers using both three-band and two-band brushes on both creams and soaps. Some people might assert that a stiffer brush might work better with soaps, whereas softer brushes seem to be better for creams. But I have not personally found this to be true. The more significant variables that affect lather quality are, in my opinion, crown size and knot density, rather than the hair itself.
Well said, I might add the massage/ exfoliating feeing of 2 banded in the face which in my opinion improves circulation while shaving.