During a partial solar eclipse, or the ending of totality, it is actually more dangerous. Basically when the sun has a significant fraction of its light blocked, your pupils dilate because the background sky dims. However the remaining exposed portion of the Sun’s surface is still very high in brightness, so if you look at the Sun at that moment, your dilated eyes get flooded with light. This is much more dangerous than looking at the fully exposed Sun, because your pupils are small enough in that situation to prevent permanent eye damage.

Total blindness from looking at the Sun under normal circumstances with the naked eye is almost unheard of, and permanent eye damage is rare. This is due to the remarkable ability of the eye to drastically reduce the amount of light that hits your retina in a bright environment. However eye damage from observing solar eclipses is more common, because if you stare at the eclipsed sun for too long, once the moon starts to move enough to re-expose the Sun, your (now dilated) eyes are unprepared.

-From the UCLA Division of Astronomy and Astrophysics.