If Yukina is the Queen of the school, Miu is its Princess Regent, bur let it never be said that Miu was ever second to the Ice Queen. In the strange battle of tradition and “progress” at Amanogawa, the respective “queens” have their own distinct realms of influence. Yukina’s is the traditional place reserved for the richest, hautiest girl at school. Miu, however stakes her claim on the modern parts with a vengeance, though she does try to remain under Yukina’s nose for some strange, inexplicable reason.
That being said, Miu is not the traditional spoiled brat believing she is deserving of reverence simply due to her birth. Miu sees the school as her kingdom, and the students as her subjects. As such, while you may earn her disdain, she takes pride and some care over what she considers hers. Most of the school looks up to Miu.
That being said, the biggest difference about Miu is her nature. She is ruthless if she sees you as a rival, and is like a Four Star General when it comes to command sense and social tactical thinking. She can raise the school against you, or make it as welcoming as a blanket from the drier with a bat of her eye.