Severity refers to the effect the bug has on the 
functionality of the application. The more the 
functionality is effected the higher will be the severity. 
For example a crash will always have the highest severity 
and a spelling mistake will have a low severity. 

Priority refers to how soon the bug should be fixed. One 
more point to note is that Severity is entered by the QA 
and the Priority to fix it is decided by the PM in 
consultation with the client relationship manager. By and 
large a high severity bug would also have high priority. 
BUT there are also exceptions to the rule depending on the 
application and release schedule. A wrong logo used on the 
home page of a company's web site is definitely HIGH 
priority whereas since its not affecting the functionality 
its Severity will be LOW. Conversely a crash during a 
roundabout scenario which is not likely to be encountered 
by the client will have HIGH severity since it has a major 
affect on the functionality whereas the PM may set the 
Priority as LOW since he might have more important bugs to 
fix which have higher visibility to the client.

The cosmetic bugs may be low priority & severity bug for
example: if there is any grammatical mistake in application
on which the customer hits very rarely since it does not
affect the functionality of apps and also the grammatical
mistake since the customer rarely hits it.