How are juvenile cases processed and how does the juvenile justice system differ from the adult system?

The short answer is, it depends. The rules generally vary from state to state.
Generally, each state has their own division of juvenile courts, separate from their regular trial (District and Superior) courts. Juvenile court cases apply to those age seventeen (17) or younger. However, certain offenses may require a juvenile to appear in the regular trial courts. The gradual trend seems to be to increase the age at which kids remain eligible for juvenile court: In 1999, 38 states set the max age at 17, ten at 16, and three at 15.
Many think that the juvenile court only deals with criminal cases involving minors (referred to as "delinquency"), however, depending on the jurisdiction, juvenile courts may also see dependency cases and status offenses involving minors as well. The procedures for each of these differs, so I'll break them down by case type.
Delinquency Offenses: These cases involve minors accused of committing a crime. Delinquency cases differ from status offenses (see below) in that the charge filed is a crime whether you are a minor or an adult. The procedures are very different from juvenile court to adult court. The biggest difference is that juvenile courts typically focus on rehabilitation and diversion programs (probation, rehabilitation, mandated therapy) rather than a juvenile-equivalent of incarceration. Juvenile attorneys generally still practice their adversarial role during motions and trial, but often work collaboratively with the state (prosecution) to find a viable remedy during sentencing to find what is really best to get the juvenile back on track. In adult court, attorneys are advocates during the entirety of the process.
Dependency Cases: These cases involve persons 17 or under that are being abused or neglected in their home. These cases typically involve a court case where the judge must decide whether it is in the best interest of the child to remove them from their parent(s) or guardian(s).
Status Offenses: these cases involve minors accused of committing an age-dependent crime. In other words, the minor is being charged with a crime that is only applicable to minors. Examples include underage drinking/possession of alcohol, curfew violations, and even skipping school.

Criminal cases in which the defendant is a minor, which means under 18 in most states, are typically tried in the juvenile justice system. Though these cases would be considered crimes in adult courts, in juvenile courts they are called “delinquent acts.” Because the goal of the system is to help the minor correct delinquent behavior and avoid a criminal record, the sentences differ from those of adults in that they generally involve rehabilitation rather than punishment. Thus, minors convicted of “delinquent acts” are far less likely than adults to receive jail time and far more likely to receive sentences that involve probation and monitoring, community service, and counseling programs. Sometimes, juveniles are placed in diversionary programs before they go to trial. Diversionary programs are sometimes called pre-trial interventions and they involve a period of probation as well as a rehabilitation program, such as one that teaches anger management or provides alcohol and drug counseling.
Minors accused of delinquent acts are tried in courts that are specifically for juveniles. The rules of conduct in these courts are more lenient than in adult courts, and the courtroom procedures are more informal. For one thing, minors are not tried by a jury, but rather, their cases are heard by a judge. This hearing is called an adjudication hearing. The judge determines whether the minor committed a delinquent act, and if so, he or she is placed in a rehabilitation program. In the juvenile justice system, the emphasis is to act in the best interest of the person accused.