What's the difference between a Clayspade and an ordinary digging or trenching bucket?

Pretty easy to see here; the Clayspade has cutaway sides, designed to minimise suction on wet or sticky material. This helps prevent clogging.

Normally, these types of soil are soft, so the Clayspade is usually not fitted with teeth. If ground penetration requires teeth, an ordinary digging bucket may be a better bet, as they are stronger and cope with higher breakout forces.