So， a teacher explained to me that 究竟 is a more formal way of saying 到底. However, I'm still curious are there any other usage differences between the two? Is 究竟 only used in examples that are relatively “夸张” Ex。 你究竟为什么要离开我。
1) "究竟" is more formal than "到底". In everyday spoken Chinese, "究竟" is rarely used. It is used almost only in drama and written Chinese.
2) Well, I would say that the magnitude of "究竟" is equal to that of "到底". They are all equivalent to "on the earth".
When used as an intensive, they are synomymous. Both translated as "on earth," "exactly."
究竟, 穷尽. literally means "investigate to the end." It can also be used as a noun which means "outcome," "final truth."
究 谋划;研究;探求. Study, investigate. Originally means "exhaust," "limit." See http://xh.5156edu.com/html3/15922.html
竟,终了，完毕. End, finish. See http://www.zdic.net/z/20/js/7ADF.htm
到底, 直到尽头. Literally means to "the bottom," "to the end."