A relief fan and an exhaust fan both remove air from a building. An exhaust fan is the term often used when the fan locally removes air that has been somehow contaminated from a specific area or areas of a building. For example, you'd find an exhaust fan connected directly to labs, kitchens, bathrooms, etc. In contrast, a relief fan removes air from the general return air coming back to the air handler (i.e. the air coming back from the whole building, including spaces like offices, hallways, etc., that could in part be reused/recirculated back into the building as long as ventilation requirements are met). The relief fans' purpose is to balance pressure within the building, essentially by removing the net airflow from the combination of air handler outdoor airflow, and exhaust airflow, and infilitration/exfiltration through the building envelope. Relief fans are common on systems that have economizers as these systems typically have high outdoor air flow rates when they are in economizing mode, which is much higher than ventilation or exhaust requirements in most cases, and so the building would be overpressurized if there was no relief fan or some other means of relief (e.g. a relief damper).
You can find a technical discussion of different types of relief systems
here, which describes the different types of layout of return and relief fan systems. As mentioned in that article, and as with almost everything in the HVAC industry, different people/organizations/regions use different terms to describe the same physical system. Some people call a relief fan an exhaust fan, especially in systems/regions that don't typically use economizers.
For the BRICK schema, I think there is value in distinguishing between a relief fan and an exhaust fan.