A good real world example of the difference between 'Categories' and 'Tags' is the way that library systems work. The Dewey Decimal System is widely used in libraries throughout the world. In this system, books are divided into 10 categories.
But within each category are further sections. For instance, within Religion, we would find sections on Christianity, Judaism, Islam, Hinduism etc. Within Languages, we would find sections on English, French, German, Mandarin and even Irish! These sections would be the equivalent of tags.