I would think it's the distinction between the "City of London" (just the city itself) and the "London Metropolitan Area" (the city and surrounding sprawl).
In fact the City of London is a tiny part of the city of London, and only one part of the centre (the north-east). It's the area that was originally defended by a wall in Roman times.
"The City" is a tiny little slice with something like one hundred residents. It bears zero relation to London, the London Urban Area, the London Metro Area, or the London Municipal Area.
