The Eggbutt cheek piece prevents pinching of the lips and gives a solid feel against the side of the face. It also prevents the bit being pulled through the mouth. The release is slower than the loose ring and also introduces light poll pressure.
The Barrel joint limits the nut cracker action, which spreads the pressure over a larger area of the tongue and lessens the pressure on the bars. The Barrel allows for independent action on each rein. Please note, this bit exerts sharper tongue pressure than the Elliptical due to the Barrel being smaller.