Can have two similar meanings in the context of marketing research:
A Brief can be a statement (usually in writing) of a business problem that could be alleviated by conducting some marketing research. Briefs are normally written by a client company for a research supplier and they usually have a background and an objectives section. A suggested methodology section can also be included.
A Briefing can be a training session prior to starting work on a study in which all specifications and details of the study are reviewed. This is generally followed by practice (or pilot) interviews where they are being used.