The Eggbutt cheekpiece prevents pinching of the lips and gives a slightly more solid feel against the side of the face. It also prevents the bit from being pulled through the mouth. The release is slower and also introduces some poll pressure.
The Waterford bit has many smooth flexible bumps or bubbles, which spreads the pressure over many points making it a light mouthpiece. The greater flexibility discourages a horse from leaning on it.
The Waterford bit is a good choice for a horse that leans, it also promotes mouthing and salivation. The Waterford is usually worn ¼" - ½" longer than your traditional mouthpiece in order to curl around the lips and maximize the effect.