From my experience ...
Bibliothek is a collection of books or an institution that administers that collection or a Bücherei.
A personal collection of books can also be called Bibliothek. But that may originate from an ancient joke, like calling a purse a treasury.
I´m a librarian´s son; I´ll ask.
, afaik, is a place where the books are stored and administered. It definitely is the narrower term.
From my mother, the librarian:
The term Bücherei first came up when artisan guilds set up libraries for their members. The term is used for non-private libraries, i.e. libraries that are run by an institution of some kind, be it open to the public or only to selected people (e.g. a Schulbücherei where only teachers and pupils of that school have access). Bücherei is also the term for the institution that runs that library, if that is its sole purpose.
Bibliothek is not only the broader term, but formerly mostly used for classic libraries, where collecting and preserving literature is the focus (so that it´s there when you need it), while Bücherei is primarily used for libraries where the focus lies on using the books.
The main difference in brief:
A Bibliothek collects books that may be needed.
A Bücherei keeps those that are needed (and dumps those that are not).
Nowadays, they are generally used as synonyms, and there are a lot of former "Städtische Bücherei" that are now a "Städtische Bibliothek".
I suppose that´s because Bibliothek sounds more sophisticated ... and people are forgetting the difference between the two. But actually, they are synonyms: one Greek, one German.