W/r/t television, it's an unrelated issue since the technology is totally different. But in film, the technology has been experimented with since the earliest days of cinema. The Technicolor process has been around since the mid 1910s, but colour in film didn't become the standard for American movies until the late sixties. There are technical, artistic, and industrial reasons for this.

Technically speaking, the early colour processes were expensive (Relative to black and white film), and they required a lot of light, which made shooting more expensive in general (More lights, more crew, bigger sound stages, and so on). Shooting in colour using those films tended to produce the oversaturated, blown out, flattened out look that we associate with films such as Singing in the Rain and The Wizard of Oz. The first Technicolor movies were musical comedies that benefited from the bright, cheery look. At this stage, colour was seen almost as a gimmick - an extra thing that was added onto a film, rather than an integral part of filmmaking. Audiences quickly came to associate colour with light entertainment; to this day, more serious films tend towards muted colour, while black and white is taken as indicative of gravitas.

The first practical colour processes, such as Technicolor, were rather expensive. In the earliest days, they also had limited availability. This contributed to the use of colour as an added extra, as studios would deliberately pick some of their films to be shot in colour. Additionally, they were only really available to major studios in major industries, so independent film, and peripheral industries, shot largely in black and white well into the sixties and seventies. This of course helped cement the idea of black and white meaning seriousness and gravitas - To the extent that American audiences were exposed to European cinema, most of it was in black and white.

The reason for the change is simply that, over time, colour processes became more practical, faster, and more refined, until Hollywood studios were outputting most of their product in colour. New Hollywood put the final nail in the coffin of the taboo against shooting "serious" movies in colour, mainly by demonstrating how developed and capable the technology was at the time. The Godfather revolutionised how American movies were photographed; if you look at the way it was shot, it's apparent that the technology available in the 30s could never have produced the same effect.