Rich Text
Formatted text fields allow the use of WYSIWYG editors. A basic configuration allows a limited set of HTML elements, which are edited via WYSIWYG tools. What you see in the editor is the rendered output, but what is stored in the database is a snippet of HTML.
Markdown is another example of rich text formatting. There are different "flavors" with support for different features.
Structured Content
Structured content is when you create content types and fields to model the content for your site. You define the fields needed to capture and represent the data on your site. For example, an event would have date and location fields.
Strategy
Two different strategies for allowing content flexibility is adding more components that allow for structured content or working with systems that give more options for embedding content.
I think the field is moving toward structured content, although there are many slick embedded content systems available. Structured content can easily be reused, but can be harder to configure since you're stringing together parts of a system. Embedded content can be more flexible for content authors, but tricky to build complex systems or enforce consistency.
Maintenance
Full HTML content can become hard to maintain. An author could use a class that is supported by the current theme. But if the theme changes, there will be no notice that the class is no longer available. Use the HTML Tag Usage module to find classes used in your content.