I can differenciate with example
suppose we identify two bugs
1. The client logo is not appearing on the web site but the
site is working fine... in this case the severity is low but
the priority is high because from company's reputation it is
most imp to resolve. After all the reputation wins more
clients and projects and hence increases revinue 

2. The Qaurterly statement event is not triggering from the
website and we are just at the begining of the new quarter
after a new release. 
In this case the Severity is High but priority cauld be low
becase we have time till the quarter ends to fix the bug

So the botemline is that in the priority v/s severity 
matrix the priority is completely bussiness impact driven
and the severity is completely technical impact driven. :)

Hope this helps !!