Consider screen readers and other accessibility technologies. If you use the h* tags as they're designed to be used, it is very easy for someone who cannot see your document to navigate within it by moving from header to header, and to navigate within sections by moving from subheader to subheader.
These are two examples of automatically extracting semantic information from an HTML document in order to enable new ways of interacting with its content. Such methods can only work if the semantic information is there to begin with. Using standard header tags in the standard fashion is a good, simple way to include such information. It is not the only way, but it is by far the most widely understood way, and you do both yourself and your readers a favor by not requiring the use of special-case exceptions to parse your content.
Of course, if your purpose is not to publish content, then this may not be a major concern for you. Perhaps you're building one of those modern-style marketing sites where the presentation is excruciatingly visual, the standard interaction methods are unnecessarily hijacked, and the content is minimally existent. In that case, I wouldn't worry so hard about it. But if you're publishing something that's got some meat to it, this is a standard worth following.
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/ .