Eine Bücherei ist eine kleinere [öffentliche] Bibliothek, a smaller (public) library. https://www.duden.de/rechtschreibung/Buecherei You would not call a huge library or a university library a Bücherei, but rather a Bibliothek. Some people having a bigger collection of books at home, would call that a Bibliothek as well.

I wouldn't use "Bücherei" for the library at an university for instance but only for a municipal library ( "Stadtbücherei"). "Bibliothek" works for both, I think.

"Bibliothek" is a collection of books available for reading, and exceptionally also for borrowing. When you have a lot of books at home, you could call this your "Bibliothek". "Bücherei" is also a collection of books, typically available for borrowing. "Leihbücherei" means lending library. In an academic context, the university library is always called "Universitätsbibliothek", probably because the Greek word Bibliothek sounds more learned and posh than the German word Bücherei.