Why would I wrap an exception if I don’t plan to have special handling? If it is exception that I didn’t throw because of a business reason, why not just throw the original exception?

If you don't have any special handling, then why are you catching it at all?

That was the "//Do stuff" part, you may want to do some cleanup (in C# usually handled by a finally block or a "using" block), send a custom notification, logging (usually I would just let the client handle logging), etc.