The main reason for using Markdown is the readability of a marked text. For instance, you can send it in a plain-text email and the reader will still understand the emphiasis, bullets, the text will be divided in paragraphs et cetera.
When you ask about storing data, it depends. If you enable Markdown in the WordPress blog engine, it stores data as the user has input it - in Markdown. In Stack Overflow, however, it seems like the data is stored as HTML. At least, the "Stack Overflow data dumps" contain HTML, not Markdown (I've seen people complainingpeople complaining) that they have to convert it back).
If you use the WMD editor, you can show the user how the outputs will look like after being converted to HTML. Even though Markdown syntax is really simple, it is not hard to make mistakes. Hence, it is best to show users the output.
Another reason for using Markdown instead of a WYSIWIG control - a WYSIWIG control allows the user to use HTML in data you are displaying on your web page. So, you have to be the one who decides when there is simply incorrect HTML and when it is an evil XSS/CSRF/whatever injection. In Markdown, you simply convert *something* to <b>something</b>
, remove any unknow HTML elements and you're done.