Brief Summary: This pilot clinical trial studies cetuximab and radiation therapy in treating patients with stage III-IV head and neck cancer. Monoclonal antibodies, such as cetuximab, may block tumor growth in different ways by targeting certain cells. Radiation therapy uses high energy x rays to kill tumor cells. Drugs used in chemotherapy, such as cisplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving cetuximab or cisplatin together with radiation therapy may kill more tumor cells.

Detailed Description: PRIMARY OBJECTIVES: I. 2 year (yr) locoregional control in cetuximab responders. SECONDARY OBJECTIVES: I. Assess secondary clinical endpoints such as the percent of patients receiving neoadjuvant cetuximab who progress by computed tomography (CT) Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria during the neoadjuvant cetuximab, the 2 yr locoregional control for non-responders to neoadjuvant cetuximab, and the complete response rate to positron emission tomography (PET)/computed tomography (CT) scan 3 months after the completion of radiation therapy for both responders and for non-responders to neoadjuvant cetuximab. II. Analyze the relationship of known deoxyribonucleic acid (DNA) mutations in tumor per the FoundationOne genomic profile, and correlate to clinical endpoints such as locoregional control. II. Analyze any changes in protein production at the tumor in response to 3 weeks of cetuximab. III. Analyze any changes in protein production at the skin in response to 3 weeks of cetuximab. IV. To investigate whether the tumor imaging characteristics including anatomical and molecular parameters evaluated by PET/CT, either alone or combined with other biomarkers can attribute to the better prediction for the clinical outcomes, as the response to neoadjuvant cetuximab; and the final clinical endpoint, the 2-year local regional controls. OUTLINE: Patients receive cetuximab intravenously (IV) over 60-120 minutes for 3 weeks. Patients then undergo external beam radiation therapy (EBRT) over 6-7 weeks. Patients achieving response continue weekly doses of cetuximab until radiation therapy is completed. Patients unable to achieve response or progression receive cisplatin IV over 1-2 hours on days 1, 22, and 43 of radiation therapy. After completion of study treatment, patients are followed up every 3 months for 2 years.