My understanding is that water flowing rapidly down the vertical surface of the dam and impacting its base rapidly erodes the face and base of the dam, quickly reducing its ability to withstand the pressure behind it (which also must be at its maximum level for the dam to be overtopped.)

For an earthen dam, erosion of the top of the dam when overtopping is also, I believe, a significant issue, but overtopping is a catastrophic failure issue for any dam.