In CW's fast lift method, you put all your effort in the bar, enough such that you give it momentum, causing it to "feel" lighter (think about how tension is almost zero right after the explosion phase of an oly lift). This means muscle tension decreases, giving them less stimulus to grow.
One alternative is to lift the bar slower, just enough to keep momentum in its place. The problem with this is that you deliberately use less muscle fibres, giving you a suboptimal stimulus.
Here's where chains come in. Since some parts of the chain are stationary on the floor while you lift it, these parts have no momentum. If you lift them off the floor, you impart momentum in them, but they also take away some momentum from you, giving you more muscular tension.
Bands have an even more profound effect. Since the bands' weight is insignificant, their momentum can be ignored, yet it still forces you down. No matter how fast you lift, momentum won't "take over" and your muscles get all the tension.
Hence, with using bands and chains, you get the best of both worlds - maximal motor recruitment and constant muscular tension. Do both.
BTW, using chains and bands are an advanced tool and plateau buster. Don't abuse and overuse them.