Pretentious statements are in italics.
I've had a few experiences with intercity bus travel both in the US and the UK, and while these encounters can't be generalized to sweeping cultural statements, my liberal arts education drives me to make sensationalist statements qualified by words like 'may' and 'might' that give plausible deniability.
The first difference is in the names: intercity buses in the UK are 'coaches'. National Express and Greyhound seem to be the major companies in the UK and the US, respectively, and they have the same system of acceptably comfortable seats and 3am stops at gas stations for snacks and smoke breaks.
In my experience, Greyhound buses are more crowded and noisy than their UK cousins. Could this be indicative of the more reserved nature of passengers in Britain?
Travelling with National Express was stressful for me because the bus driver did not announce the names of the multiple stops in each town or even the names of the towns we were stopping in. Since I usually travel at night, it was very difficult to tell when my stop was or if I had missed it. This lack of announced information could be rooted in a culture of unwritten rules and traditions.
I should probably say that I've enjoyed travelling on buses in both countries, and, as one might expect, the experience improves with experience.