Because the twins were taught to hate her. The prince wasn't told no as much as she should have been so he is used to having is way and paying little to no consequence for it while his full sister has to uphold female modesty, or she would be exactly like him.
Because of their station, they aren't going to change. They see themselves above everyone else and it has been enforced.
Because her mother encouraged her to fight back after practically forgetting her in a corner, our MC has been taught to return such scorn tenfold to avoid it happening again; but she can't do that with them so it's constant tension.
We can see how kind she could have been, born to someone else at a different station in life; her attitude is a defensive mechanism. We don't have the same for the twins, only the knowledge that others enforced this kind of behavior. The prince especially is what happens when a parent thinks a child's rebellious behavior is "cute" and as they get older, they wonder why they get into so much trouble.
I understand that the twins sucks, but I dont get the whole plot going here, cause she is not a ilegitmate child nor a daughter born out dirty acts, the king had the right to have a concubine, I get her being not aknowleged as much as the twins or not having the right to the throne, but dont see the reason for all that mistreat in her childhood or the behavior of the people around her know that her mother was made a Queen, she has the right to be upset ant throw tantrus as much as she likes. Why her little brother who has the same blood as her get all the good things and she has to go through those hardships? What I want to mean is that someone is responsible for the servants behavior in her childhood, someone started the whole outcast situation on her for some twisted purpose and I think it was or her mother deeds or the late Queens

I just dont get why they mistreat her for being a ilegitmate child, she isnt even a child of a wedlock, her mother was a concubine and turned out to be the queen after the previous died, why blame her for that, she is indeed the king and queen daughter