The Fallout universe is intended to be a purely fictional representation of what people in the 1950s thought the future was going to look like. As such, there's no "explanation" for why culture stayed that way other than that's how people in the 50's lived:
The Fallout world is much like Torg - physics and natural laws are not the same as in our universe, but are based instead on 50s sensibilities and pulp era comics - the Fallout universe is what people in the 50s believed the future would be (with a lot of nuclear warheads dropped on it).
(The Fallout Bible, Part 9.)
The Fallout setting was heavily influenced by a 1950's science fiction anthology called the Worlds of Tomorrow.
There were two major differences between the history of the "real world" and the Fallout universe, one of which you've already hit on:
- In the Fallout universe, the transistor was never discovered, and
- Atomic fusion, particularly portable atomic fusion, was perfected.
Pre-war Fallout is a vision of what would have happened without the miniaturization allowed by transistors, meaning most of the electronic advances of the 70s forward could not have happened. Would we have the same kind of music if there were no synthesizers, turntables, guitar effects pedals, CDs, iPods, etc? What would cars look like if everything running them was still big and mechnical? Fallout is Black Isle's attempt to guess what that might be.
Post-war Fallout is essentially what people during the "Red Scare" thought the world would be like if there was a nuclear war in the 1950s, it just didn't happen in-universe until 2077.