Why do mothers not care about the male child's 'awrah (private parts) being exposed, while they rush to cover the female child's 'awrah? And is the female child's 'awrah more important than the male child's 'awrah in Islamic law?

There is no Shari ruling for the awrah of an un-discerning male or female child. Rather, the ruling was revealed for discerning children (7-10 years old), where the awrah of a male is the front and back private parts, and the awrah of a female is from the navel to the knee. Some societies may find the uncovering of an un-discerning female's awrah more objectionable than that of a male, based on what her awrah will become after the age of discernment.