I can't speak for the UK but in the US, I'm not sure I have ever heard anyone use the term "to bring off". It exists, and there appear to be nuances, but I've actually never heard it used.
Having said that, I did search a bit online and the difference between them seems to me the implicit difficulty. To pull something off tends to defy a greater expectation of failure. Where "to bring off" might simply mean to accomplish something, "to pull off" seems to mean "did you surprise everyone with your accomplishment".