"The libraries are old."
Translation:Die Bibliotheken sind alt.
There isn't really one, you can use them both. One of them will be used more depending on where in Germany/Austria/Switzerland/Luxemburg/ you are, but you will always be understood.
Talking about a university library, a national library, ... however, you mostly use Bibliothek. And if you have a home library, that's also a Bibliothek.
In my opinion you use Bibliothek / Bibliotheken for the more important ones with plenty of volumes (Landesbibliothek, Uni, Museum, ...) and Bücherei for the local ones with a very limited stock (Stadtbücherei, Schulbücherei). I would not go so far to use these terms for my own collection of books, especially since I do not lend them for a living. I would call it "Büchersammlung" at best.
I corrected it, I have no idea why I wrote Bibliotheke. Thanks!
With a home library, I mean really a room with only bookcases and maybe a comfortable chair to read in, not so much the collection of books. A lot of my (a bit more wealthy and fancier than I am) friends have a room like that, but noone would call it their Bücherei, it's the Bibliothek. I have a rather large collection of books too, but, as they can be found all over my place, I indeed call them my Büchersammlung. Or my Precious, but that's just me ;)