The Eggbutt cheekpiece prevents pinching of the lips and give a slightly more solid feel against the side of the face. It also prevents the bit being pulled through the mouth. The release is slower and also introduces some poll pressure.
The Taylor Chain bit is has an action between a Waterford and a Snaffle. The Short length of twisted chain allows it to take the shape of the tongue and as a result distributing the pressure across the tongue like a Waterford. The ends the mouth piece act more like a snaffle exerting pressure on the bars.