A lot of it is semantics. A bunch of HTML5 is semantic vs. actually having different behavior. <header> behaves no differently than <div id="header"> but is more semantically correct.
A simpler way to say "more semantically correct" is "correct." Please don't trivialize the importance of semantics. Proper semantic representation makes documents more accessible.
I don't find the header, article, section, etc tags to be sufficiently clear that they are always used in the same way by different people. But maybe I'm missing something -- how do they improve accessibility?
Take a look at http://accessiblehtmlheadings.com/ for a pretty solid introduction. The tl;dr: is that, used properly, they enable screen readers and similar to identify document structure and make it more easily navigable.
For a considerably more detailed treatment with special attention to concerns around accessibility and some information on other document elements with similar significance, take a look at http://accessiblehtmlheadings.com/ .