What are the differences between neosporin with or without bacitracin zinc?

None: They are both useless and sometimes harmful. Throw them away. Studies on people with surgical wounds demonstrate clearly that application of vaseline or Aquaphor to wounds is superior to using aminoglycocide or polypeptide antibiotic ointments. The purveyors of these useless products should be ashamed of taking advantage of so many misled suckers.