By which I mean, how is a “Justice Court” different from any other court?
What is the context? It’s what some western states call their lowest trial courts - the ones with jurisdiction over suits in which small matters or amounts of money are at issue. East of the Mississippi they’re called county courts (or possibly magistrate courts, depending on how the jurisdiction arranges such things.)
Does OP live in New York State? There, a justice court is a locally- funded tribunal in a very rural, low-population area, also called a Town or Village Court. There are more than 1200 such bodies and until recently operated with little oversight. Needless to say, there have been criticisms of this system which lead reform beginning in 2007.
Arizona and Nevada also have Justice Courts as the lowest rank of the judiciary, dealing with petty offenses, small claims, and in some cases warrants and inprocessing of cases to be referred to a general court of first instance.
The name comes from being presided by a Justice of the Peace who is not necessarily a licensed lawyer.