Graduate and postgraduate seem to be fairly interchangeable terms, at least to many of us, when referring to courses taken after your first or undergraduate degree, but the different terminology is not just a case of American English vs British English.
So, let’s try and make the differences (or indeed similarities) a bit easier to understand…
What You're Studying
There are some institutions – in both the US and the UK – who define a graduate course as being one that requires an undergraduate degree to gain entry to and results in a qualification that can also be gained at the undergraduate level. A postgraduate course would be for further study in greater detail of subjects already studied at the undergraduate level. For example, a graduate degree in Physiotherapy would be the masters course that results in a student becoming a qualified Physiotherapist and a postgraduate Physiotherapy course would be a masters course for qualified Physiotherapists. A conversion course to gain an LLB or law degree in the UK would require an undergraduate degree to gain entry but would result in a degree that could have been gained studying a longer undergraduate degree program and this would be a graduate course. A Masters of Laws usually requires an undergraduate law degree and would be a closer look into specific areas of the law, so is considered a postgraduate degree course. Further study, such as a PhD or Doctoral studies are referred to as both a graduate and postgraduate studies depending on whether you are in the US or the UK.
Location, Location, Location
In the US graduate courses are studied at Graduate School, which are universities that may or may not offer undergraduate degrees as well. In the UK it is unusual to find an institution that only offers postgraduate courses and they tend to be small colleges offering a limited number of courses around a single subject.
Duration Of Course
Generally, a graduate or a postgraduate course, like a masters program, in the US will last around two years. In the UK a postgraduate course, like a masters, will most likely last around one year both with full-time study and a graduate course, like a masters to retrain, would last two years. A PhD will last three to seven years regardless of where you are studying.
There are institutions that would define a graduate course as one that just requires an undergraduate degree to gain access to rather than one that requires additional qualifications, such as honours with your undergraduate degree or significant work experience. Under this definition a Postgraduate Diploma would often be a graduate course and the following masters course would be a postgraduate course.
There are many similarities between graduate and postgraduate courses. Both require research and consideration before applying and undertaking the course as both have a significant amount of personal research and study compared with an undergraduate degree. This means that they are a significant investment into your future career regardless of what you call them, and since different institutions use different terms don't worry too much about what you call the course. It's best to stick to the terms that the university you are applying to or attending uses to avoid confusion.