To keep silent is the more common phrase, and means the person doesn't speak, especially if there is some secret involved.
"He knew who had caused the damage to the car but kept silent."
"The police asked for more information but he kept silent until his lawyer arrived."
I only know of "keeping silence" in a religious context. Monks and nuns who live in groups and do not speak "keep silence", they don't interrupt the silence of their living areas. They "preserve" the silence.
"Keeping silence is important for the nuns so that they can concentrate on their prayers."