What is the function of the high court?
The High Court is the highest court in the Australian judicial system. It was established in 1901 by Section 71 of the Constitution. The functions of the High Court are to interpret and apply the law of Australia; to decide cases of special federal significance including challenges to the constitutional validity of laws and to hear appeals, by special leave, from Federal, State and Territory courts.
There is no difference. The supreme court is called the High court because it is the last court in which federal questions can be decided. There is no "higher" court that can review a case decided by the supreme court. In the United States, the "High Court" is a colloquialism for the US Supreme Court. They are one and the same.