According to the New Jersey Driver's Manual if a student driver commits a traffic offense, responsibility will be with both student and instructor. In addition, holders of GDL Permits or Provisional licenses may receive a $100 fine for violating any GDL conditions.
All holders of a Provisional Driver License whose provisional licensing period is NOT extended by NJ Motor Vehicle Commission beyond the standard 12 months MUST upgrade to a Basic Driver License after the end of those 12 months. Motorists eligible for, but WHO DO NOT upgrade to, a Basic Driver License at the end of those 12 months will remain subject to MVC's Provisional Driver License regulations and could be cited by law enforcement for violating Provisional Driver License regulations.
The NJ driver's manual does not note if a provisional driver must commit another traffic offense (a primary one) to be stopped by law enforcement and cited for violating a restriction of your provisional, try contacting the New Jersey MVC to find out more about this topic.
As for insurance, depending upon what is placed on your driving record, it may have an affect on your insurance rates. Since insurance companies' rating systems can differ, discuss this issue with your insurance agent.