Question: So, a teacher explained to me that 究竟 is a more formal way of saying 到底. However, I'm still curious if there are there any other usage differences between the two? Is 究竟 only used in examples that are relatively “夸张”

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> E.g. 你究竟为什么要离开我。
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Answer: 
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".
1 When used as adverbs

Both 到底 and 究竟 can be used as adverbs, and they are nearly the same when they are used to emphasize or to express strong emotion(normally negative ones). In this situation, the only difference is 究竟 is more formal than 到底. Here are a few examples.

你究竟去不去？

他们究竟是谁？

你们到底想干什么？

他们到底是怎么考虑的？

When used as an adverb, 到底 can express the meaning of till the end, finally. 究竟 does not have this usage. But we use 最后 or 终于 more often. Some people use 到底 more often, so you can see this usage as a personal style.Here are a few example.

他们到底去了欧洲。They went to Europe finally.

你们到底还是不相信我。They did not trust me till the end.

我们公司到底把你们请来了。Our finally successfully invited you in the end.

2 other differences.

究竟 can be used as a noun, it means result, reason, things that happened. A few examples.

我们都想了解一下究竟。 All of us want to know what happened.

无论什么问题，他都会问个究竟。 Whatever the question is, he asks everything.