The difference between Mage:getSingleton() and Mage::getModel() is that the former one does not create an object if the object for same class is already created, while the later creates new objects every time for the class when it’s called.
Mage::getSingleton() uses the “singleton design pattern” of PHP. If the object is not created, it will create it.
Mage::getSingleton() is mostly used when you want to create an object once, modify it and later fetch from it. Popular example is session, you first create a session object, and then add/remove values from session across different pages, so that it retains your values (e.g. cart values, logged in customer details, etc.) and doesn’t create new session object losing your last changes.
Mage::getModel() is used when you want to have the fresh data from the database. Example is when you want to show records from database.