distinctions between running an IIS web site as Mr. X and running the application pool as Mr. X

...well, one of them is that the web site will try to access C:\Windows\Microsoft.NET\ and it's files and need to be able to modify them (assuming an ASP.NET app) and the application pool's user will govern which user attempts to access a database when attempting to use a connection string that specifies integrated security in lieu of a specific, hard-coded user.