The Eggbutt cheek piece 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 Elliptical is a 45 mm lozenge, it is stronger than the Control Plate due to it having a smaller surface area. Consequently, a horse using his tongue as a means of pushing against the bit and therefore taking control, will draw back and tuck his chin in, coming back to a better point of control.