To my understanding, they both refer to circular symmetry. I know U(1) is the complex plane, and I think O(2) is real numbers. I have seen them used more or less interchangably, but is there actually some subtle difference or are they just two ways to say the same thing?
I'm specifically interested about this question as it applies to phase transitions in materials, specifically with BKT-like transitions. It seems people use U(1) and O(2) interchangeably to describe a BKT transition.