Pevsner was a Modernist. Modernism was the main architectural orthodoxy from about 1920 until the Second World War. It was meant to be in tune with the modern age and it was inspired by the machine and by industry. Modernist designs were meant to be efficient and supremely rational.
As an advocate of Modernism, Pevsner was concerned that early Modernist buildings should receive the same protection as historical buildings. So he suggested in 1970 that a very selective group of twentieth century buildings should be conserved - mostly iconic Modernist designs as you would expect. They're now known as the "Pevsner 50."
The De La Warr Pavilion at Bexhill-on-Sea is a prime example of British Modernism, designed by Serge Chermayeff, who was English despite his name. It has clean lines, pristine white elevations, and no superfluous decoration. The centre point has these cantilevered planes which express their construction.
The Twentieth Century Society originally followed Pevsner's selection process and his taste for Modernism, but has since broadened its remit. It now differs from Pevsner, because as well as valuing Modernist buildings, it values things Pevsner would regard as "historicist", too backward-looking. For example, Gothic churches were still being built in the 1960s. But the Twentieth Century Society campaigns for listing regardless of style.
The Twentieth Century Society is organized by Alan Powers and Gavin Stamp. It aims to "educate and inform taste in Britain." So it tries to spread the word on modern architecture to get the public to value buildings. One could argue that this is quite didactic, telling people what they should like. It assumes that the public are hostile towards modern buildings because they don't understand them, which is quite an arrogant assumption. But on the plus side it recognises that the public is really the only force that can protect our architectural heritage.
They got the famous red telephone boxes listed, which were designed by Sir Giles Gilbert Scott, who was also the architect of the Bankside Power Station on the Thames. The Twentieth Century Society campaigned to have this building listed, but the Tory government of the time refused. This was the era of Thatcher's return to Victorian values and support of the heritage industry. Radical 20th century buildings were out of favour. But thanks partly to the Society's support it was reclaimed and converted into Tate Modern.
The Twentieth Century Society's website states that, "prejudice and taste in architecture and design is fickle". Again the statement is a little patronising, it implies that the public respond to architecture out of prejudice and ignorance, not true taste. But it's true that Modernism was pretty much rejected by the public. Much post-war architecture was seen as a complete failure. One example is Ronan Point, which was a utopian Modernist development that failed spectacularly. A gas explosion caused several floors to collapse.
By the 1960s, Modernism had developed into the Brutalist style, which was hugely unpopular with the public. It was promoted by the architects themselves and critics like Reyner Banham.
An example of Brutalism is Gateshead carpark (1967). It's threatened with demolition. The council and most of the public want it demolished. But the demolition is opposed by fans of the film Get Carter, in which it appears. It was designed by Owen Luder, who was a president of RIBA.