They both basically mean the same thing.
She refuses to acknowledge the fact that her son was smoking.
.
She closed her eyes to the fact that her son was smoking
She shut her eyes to the fact that her son was smoking.
The first one uses visual imagery to indicate that when she sees her son smoking, she shuts her eyes, preventing her from seeing it, so she doesn't know he's smoking. Of course, this is silly, as she needs to see it to shut her eyes to protect her belief that he's not smoking.
She turned a blind eye to the fact that her son was smoking.
This phrase has a bit of history behind it. "Turned a blind eye" might better be written "looked at it with unseeing eyes." There is a old naval legend (perhaps real, perhaps not?) of an Admiral that saw the signal to retreat, but believed he could win the battle. So he exclaimed to his Commandant who mentioned the retreat signal that clearly the other ship was signaling something, but he couldn't see it due to his "blind" eye. An Admiral outranks a Commandant, so the ship remained fighting.
In the modern use, the eye isn't considered blind, but rather the observer chooses to deliberately ignore the evidence. In this case, the mother's belief that her son couldn't be smoking leads her to ignore her observation that he is smoking.