Lucas nailed the most important aspect. Let me explain this in a slightly different way:
In the browser the doctype serves exactly one purpose, to set the rendering mode: Quirks, almost standards and full standards mode. This used to be simple until MSIE 8 introduced a second switch, "compatibility mode" = MSIE7 bugs and limitations are preserved on purpose.
An HTML5 doctype will (usually) override that setting, and is thus somewhat more powerful than XHTML 1.x or HTML 4.01 doctypes. (MSIE may override it sometimes, though.)
The definitive guide to doctype switching and layout mode is at: http://hsivonen.iki.fi/doctype/
The second thing that will happen is that you get your pages validated according to the HTML5 rules. Some elements and attributes that were allowed in HTML 4 are gone and a bunch of new ones have come. A few syntactic rules have come as well. E.g. this is illegal in HTML5:
<div>
<p>foo</p>
<span>bar</span>
</div>
Rule: You must not mix block and inline elements as siblings in HTML5.
The semantics and syntax for HTML5 is mostly like the one in HTML 4, so most well written HTML 4 sites will continue to validate. The main difference is that you mat start to use the new stuff and still be valid.