This seemed like a good way of thinking about the difference between categories and tags, so I ran with this:
Once upon a time, WordPress.com only provided a Category option.
Categories allowed for a broad grouping of post topics, but when you wanted to describe a post in more specific terms, more categories were required. That led to very long category lists inside the blog and very long lists in Categories Widgets.
So we now have tags, too.
Tags are similar to categories, but they are generally used to describe your post in more detail.