Severity tells the seriousness/depth of the bug where as
Priority tells which bug should rectify first..

Severity->Application point of view
Priority->User point of view
-------------------------------------------
Some bugs might have high proirity,but according to 
application point of view,that bugs might be of low 
priority.
Ex:GUI bugs->Font,Alignment,spacing...

If a bug is severity means ,that bug will affect the entire 
application.Can be a show stopper

severity tells  impact of the bug on the application.
priority tells urgency of Bug Fix.

Priority is that how serious is the bug in case of Company
point of view

Sevierty is that how serious is the bug in case of
Technical/Application point of view

Severity is related to priority I can explain this with
example say a bug is a show stopper<say application crushes
down> then its 
Severity would be high <ie imact to application is high>
Priority would be High(1) <should be rectified as soon as
possible>

Rightly said sevierty is impact and priority decides how 
soon it should be fixed...
genrally when sevierty is high priroty is also high but
i will like to give example where severity is low and 
priority is high.

If the title bar of any website is wrriten incorrectly it 
doesnt make any difference to the funtionality (low 
sevierty)of the website but it has a high priority to be 
rectified.

Based on the severity, priority is assigned by the developer.

Severity: the seriousness of the defect in terms of 
functionality.

Priority : the importence of the defect in terms of customer