Comparing Two Different Myeloablation Therapies in Treating Young Patients Who Are Undergoing a Stem Cell Transplant for High-Risk Neuroblastoma

Inclusion Criteria: * Diagnosis of neuroblastoma or ganglioneuroblastoma by histology or as evidenced by the presence of clumps of tumor cells in bone marrow and elevated catecholamine metabolites in urine meeting any of the following criteria: * Patients with newly diagnosed neuroblastoma with International Neuroblastoma Staging System (INSS) stage 4 disease are eligible with the following: * MYCN amplification (i.e., greater than four-fold increase in MYCN signals as compared to reference signals), regardless of age or additional biologic features * Age \> 18 months (i.e., \> 547 days) regardless of biologic features * Age 12-18 months (i.e., 365-547 days) with none of the following three favorable biologic features (i.e., non-amplified MYCN, favorable pathology, and deoxyribonucleic acid \[DNA\] index \> 1) * Patients with newly diagnosed neuroblastoma with INSS stage 3 are eligible with the following: * MYCN amplification (i.e., greater than four-fold increase in MYCN signals as compared to reference signals), regardless of age or additional biologic features * Age \> 18 months (i.e., \> 547 days) with unfavorable pathology, regardless of MYCN status * Patients with newly diagnosed INSS stage 2a or 2b with MYCN amplification (i.e., greater than four-fold increase in MYCN signals as compared to reference signals), regardless of age or additional biologic features * Patients with newly diagnosed INSS stage 4s with MYCN amplification (i.e., gr…