There are a few.
The UK is a parliamentary system, meaning their head of government comes from their legislature, while the US is a presidential one, meaning we elect ours separately.
The UK has a hereditary upper chamber.
The UK has a living constitution, while the US has a written one.
The UK has a monarchy, meaning the King is their head of state, while the US does not, meaning the President is ours.
There are more, but I like to focus on a thing that unites us, and that is that we are the two major nations of the world that are not one political state divided into pieces. Instead, we are a collection of (four or fifty) political states, each of which gave up part of their power to operate together as a union. It may seem minor, but it is responsible (at least in part) for the fundamentally different way we operate and think of ourselves, compared to the rest of the world.