WESTUPLACE has a good idea concerning operating voltage level.
Also, when power is cycled at the dishwasher, is it done from the controls on the dishwater, or a separate disconnect?
If it is by controls on the dishwasher, it is possible that the on/off switch does not turn off everything. This is similar to modern TV's with remotes. When the TV is turned off via the remote, for example, or from the on/off switch on the TV, internal monitoring circuits are still energized. Could this be the same with the dishwasher? If so, this could explain why cycling power from the CB has a different effect than when cycled from the dishwasher. Cycling the CB resets and reinitializes the dishwashers monitoring/control circuits.
This explanation still doesn't answer your main question though of why the washer doesn't work right, which I can't help you with.
I think WESTUPLACE still has a most probable answer. Check the washer to see if it has reconnection configuration for 208VAC.