Does magic make it easier for a competent wizard librarian to search for information, compared to what a competent Muggle librarian can do? Can you give examples of what he can do with magic?
The main point here is looking for information. Wizards don't have computers (at least at the time of the events in the books), but they may have some magical help I haven't noticed. Magic used for other things in a library, such as books that scream when stolen, or magically reinforced buildings, aren't on topic for this question.
I can only think of one example from the books, and even that is doubtful: Hermione has somehow Summoned the books about Horcruxes from Professor Dumbledore's room in Deathly Hallows.
We don't see much of Madam Pince's work. I don't recall any time the students ask for her services. I don't know if that's because they don't trust her, or because we only see them look for information about topics they would rather keep secret (e.g. the Philosopher's Stone, Slytherin's monster, or Horcruxes). There would certainly be a good bit of use for her help, given the amount of essays and other coursework the students do, and the lots of research Hermione does.
Hermione certainly seems to do a lot of searching in books all by hand, and I don't recall her using any magic for this. But it is possible that it happens off screen, or that this just isn't a strong suit of Hermione, and other wizards are more competent. It is also possible that Arithmancy is relevant, and we know very little about that subject.
Remark about why I'm asking: Given that I work with computers, I always sort of wanted to know how wizards in Harry Potter can replace them (at least partly). Magical fantasy fiction never seems to spend much time about information processing, possibly because it's so new and the books I'm reading are old. This question asks only about a specific area of this. Part of what made me ask right now is my previous question What is Divination, as practiced by Wizards, really like?