What is the difference between "graduate students" and "postgraduate students"?

There isn't an entirely agreed distinction between "graduate" programs and "post-graduate" programs -- the terms are frequently used interchangeably, especially in academia. "Graduate" is used as shorthand for "Postgraduate" in the academic setting.

If someone has finished a higher degree and is continuing their studies they would still generally claim to be a "graduate student," rather than a "postgraduate student"; the same term that someone with only an undergraduate degree would likely use - it's the course they're on at the time that matters.

Outside the academic setting, "graduate" tends to mean someone with either an undergraduate or higher degree (or both). "Postgraduate" suggests someone on a course of study (where typically you need to be a graduate student to apply).

Extra:

Having looked at my Uni's website, it mixes the terms fairly interchangeably. The first and most obvious example of this is that if you click on "graduate courses" under the Admissions section of the home page (http://ox.ac.uk) the breadcrumb navigation trail shows "Home > Admissions > Postgraduate courses".

Cambridge is a lot more fussy, and delineates between graduate and postgraduate.: http://www.ice.cam.ac.uk/faqs/3-credit-faqs/140-what-do-undergraduate-and-postgraduate-mean

In New Zealand universities, a "post-graduate" student is one earning an advanced degree (masters, doctorate, diploma) in the subject of their undergraduate degree. A "graduate" student has also earned an undergraduate degree but is continuing study in a subject for which they have not explicitly earned a bachelor's degree.