The waterfall method uses a single piece of carpet for the staircase, draped over the stair tread (horizontal part of the step) and secured where the tread and riser (vertical part of the step) meet. Cap and Band installation, mostly used by experts, uses individual pieces of carpet for each step and additionally tacks the carpet tightly to the underside of each tread.    

Cap and band	Waterfall

Many homeowners prefer the traditional "finished" look of cap and band installation. Additionally, carpeting on a single step can be replaced if needed (i.e. stained), without having to re-carpet the entire staircase. And, the waterfall installation can come loose or sag as compared with cap and band but know that most installers will charge more for cap and band installation of your stair carpeting.