Based on my research, I can posit what I believe to be a satisfactory answer in terms of US terminology. The US refers to the U.S. supreme court members as "justices" all the way back to the Judiciary Act of 1789. The more salient point is in Article II. section 2 of the US Constitution gives the presidency the uncontested authority to appoint the "Judges of the supreme court".
Now I had come across several sites which suggested that the difference between a judge and a justice (not necessarily limited to the US) was that the former required a law degree and the latter did not. The U.S. Bureau of Labor Statistics Occupational Outlook Handbook 2010-2011 had the following to say about Judges, Magistrates, and Other Judicial Workers.
A bachelor's degree and work experience are the minimum requirements for a judgeship or magistrate position, but most workers have law degrees and some are elected; training requirements for arbitrators, mediators, and conciliators vary.
Essentially, while the US constitution grants the president to pick anyone they damn well choose regardless of qualification to the supreme court, judgeships seem to have more requirements on paper. Granted the high level of scrutiny of supreme court appointments and the presidential advantage of appointing a lifetime sitting justice has kept ham sandwiches from being appointed (regardless of one might think of Antonin Scalia or Clarence Thomas, they certainly are not sandwiches).
Based on the fact that the US judicial system was modeled on the English common law system, the terminology almost certainly came from there and is likely to have a similar meaning. Although I couldn't find any etymological evidence to this effect, I certainly wouldn't be surprised if the terminology came from the relationship between the "justices" and the crown. While all magistrates in English common law of the early American period all served at the behest of the crown, the American relationship and the more lofty title seem to suggest a higher level of prestige awarded to a justice as opposed to a judge.