That’s a great question. I’d need to look it up to give a more precise answer, which might or might not end up being an “ELI5” answer. But it seems like a semicolon works better in that spot because there is more of a pivot than a continuation.

I can see how we might view the second clause as a continuation: it does give more information. His inability to find the book is the reason why he spent so long in the library.

But it really seems like more of a contrasting situation. If he spent that long in the library, we would expect him to find the book that he wanted; however, he didn’t.

It seems like a semicolon works well in your example because we can so easily slide the word “however” into that sentence: “Sandip spent three hours in the library; however, he couldn’t find the book he wanted.”

If you think that a colon conveys the meaning that you’re aiming for, the word “because” probably conveys that even more clearly: “Sandip spent three hours in the library because he couldn’t find the book he wanted.”

I use a mnemonic based on the first letter of each word: colon = continue and semicolon = stop.

The clause after a colon will continue the idea presented in the earlier clause. The clause after the colon gives us more information in some way. In your example, the promotion is more information about “what he worked for.”

A semicolon serves as a stop. It separates two ideas that are related, just as any two sentences in a paragraph would be, but are different or contrasting in some way. As u/ikantolol wrote, it works where a period seems too strong and a comma seems too weak.

Semicolons are also used to separate items in a list when those items contain commas: “I had fun rewatching Star Wars Episode IV with my aunt, a screenwriter; her husband, an astrophysicist; and their old friend from high school, Mark Hamill.”

As is always the case in English, there are exceptions and other uses, but for me, “colon = continue / semicolon = stop” has been a good guideline.