Difference between Mage::app() and Mage::getModel() in magento.
Mage::app()
Mage::app() function is used to bootstrap your Magento application (setting up configuration, autoloading etc) and is useful when wanting to access Magento models.It is a static method so it can also be called at any point in your application to return the current Mage_Core_Model_App object which you can use for example to get current configuration values e.g. Mage::app()->getStore() will return the current Mage_Core_Model_Store store object.
Mage::app() is similar to Mage::run() found in your index.php file. The difference is that Mage::run() will amongst other things also invoke the MVC, routing etc part of your application as per default and control the request/response directing you to a page and instantiating the blocks and layout template rendering.
Mage::getModel()
Mage::getModel() is simply a factory method that will return a new instance of a class based on the class alias you provide. For example Mage::getModel(‘customer/customer’) will return a new Mage_Customer_Model_Customer object. If you want the same object back each time throughout your request you can use Mage::getSingleton() instead.