With closed eyes it is possible to fall asleep and with fully opened eyes it is possible to get connected with the surroundings.

Another thing: the fully opened eyes are the one extreme and closed eyes the other. Both known as Yang and Yin. Yang for fully opened eyes and Yin for closed eyes.

Both Yang and Yin have their qualities just exact opposite to each other, for instance, Yang is active and Yin is passive. The half closed eyes help in balancing where both Yang and Yin are 50% and 50%, and one neither feels sleepy nor gets connected with the surrounding.

The great example of the half closed eyes is the Shiva, the Mahadeva. It is very hard to find any picture of him without the half closed eyes, as he is considered the greatest Yogi as well.