A quoted company physically trades its stock on a Stock Exchange (e.g. London Stock Exchange or AIM).
A publicly listed company is a legal status at Companies House and can be trading without being listed on a stock exchange. Legally, members of the public can still buy stock of a publicly listed companies but would do so via other means (e.g. a broker).
This does not apply in the US and quoted and publicly listed company means that they trade on a stock exchange.
GBBE, GBBUK, UKBB