What is the difference between a Case Report and a Brief Report? What exactly is the difference between a Case Report and a Brief Report in the New England Journal of Medicine?
Short answer
In NEJM, case reports are called brief reports.

Background
A case report is in general a report on a single subject/patient. It is a broadly used term in many journals. From the wiki page: 


  [A] case report is a detailed report of the symptoms, signs, diagnosis, treatment, and follow-up of an individual patient. Case reports may contain a demographic profile of the patient, but usually describe an unusual or novel occurrence.


The NEJM in specific does not feature an article layout for case reports in particular; instead, they have devised a common outline for those papers that either report on one case (case reports), or on a bunch of related cases, or of one specific family. From the Author's instructions of NEJM:


  CLINICAL CASES
  Brief Reports usually describe one to three patients or a single family.


Hence, in the NEJM, a 'case report' is one of the article outlines under the header 'brief report'.