If I understand your question correctly, you are asking what the difference would be to call the Azure table code from within a local application as opposed to your web application. There is no difference at all. Azure table storage is simply an Internet based storage solution that is accessible from any Internet connected device. This can be an Azure application, a seperate website elsewhere or a stand-along application on your own computer.
As long as you have the correct keys to access the storage inside your Azure connection string in your application, or parts of the storage have been made publicly accessible, you will be able to access your Azure storage account. This is discussed more fully in this Azure table storage
] discussion if it is any help.
Give me a shout if I've not explained this properly or if I've misunderstood your question and I can clarify things further.
Hope this helps.