It strikes me as quite strange to have one entrance and then one library on the ground floor and a library with a different name on the floor above, but I realise there’s a history there.
The Guildhall Library’s website explains it best:
In 2009-2010 the Library was transformed again with the former Printed Books section forming the nucleus of the Library you see today. The Prints and Maps and Manuscripts sections moved to London Metropolitan Archives but some major manuscript collections are still housed at Guildhall. Guildhall Library now shares a building with the City Business Library, so users can now move very readily between current and historical business resources as well as having access to the Internet and the Department’s extensive range of online resources. In fact the City Business Library, once called the Commercial Reference Room, has returned to its original home: first housed at Guildhall Library it moved to Basinghall Street in 1970 and later to Brewers’ Hall Garden.
The new Guildhall Library is a major public reference library, very connected with its past, holding a wide range of important works and sources including: a comprehensive collection of printed books on the City of London and its history, the Lloyds Marine Collection, a large collection of pamphlets from the 17th – 19th centuries covering political and social issues, a complete run of the London Gazette from 1665 to the present, extensive parliamentary resources including eighteenth-century poll books and a complete set of House of Commons papers from 1740, broadsides and an unrivalled collection of local and trade directories from 1677 to the present.
They’ve got a good blog at about their exhibitions and events with pictures: https://guildhalllibrarynewsletter.wordpress.com/