Priority indicates how soon the bug should be fixed. Severity indicates the seriousness of the defect on the product functionality. More severe the defect is, higher the priority is given to ensure the defects are fixed as soon as possible to maintain the quality of the application.
Example: A bug that would damage data is more severe than something that is just annoying like say some functionality taking too long to load. Both should be fixed but those with higher negative impact should be fixed first.