Severity is based upon how much of the application is affected. Are all pages broken, is it important? This is an assessment of the issues extent without dealing with where exactly it happens. Also this is a discussion of how severe the problem is without regard to where it falls on the ToDo list.
Severity is usually set by the tester when the defect is found.
Priority is a general assessment of the problem and where it should be inserted in the to-do list for the team. Priority addresses the when. When should we fix this? Should it be handled early in the cycle or late?
Priority is often set as a part of defect triage. Determining how severe it is may also be a part of development, or business, triage.
Of course, this example is based on a 4 point scale*, if you have more levels, just redistribute them. The big thing here is that the whole team has a common understanding of how and why certain priority and severity assignments are made, and that they all adhere to that understanding. If the team is cohesive on how these are used it will make the process flow smoother.
*SpiraTeam comes with a 4-point scale for both priority and severity deliberately. If you use a five point scale (or any odd number), you will find during triage that everyone will want to assign all the incidents to priority / severity 3 (right in the middle) and you won't have a good way of understanding which ones are the more important. By forcing a priority / severity 2 vs. 3 the triage forces the 'hard' decision early.
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