Using properly constructed HTML5 will help your rankings, and improve your site's accessibility. Given identical semantics (in a prefect world), it will rank no better or worse based solely on semantics than an HTML4 or XHTML1 site. However, using HTML5 properly will improve your semantics simply by the nature of its design.
Google has put a lot of money and engineering effort into HTML5, including launching the site HTML5Rocks.com. The more semantically correct your code is, the better your rankings will be for the subject of your site.
Additionally, if properly constructed, HTML5 makes your site faster to render and generally smaller to download, improving page speed (which was recently added to Google's ranking algorithm).
<aside> (haha HTML5 humor) Google has for quite a while supported multiple
<h1> tags on a page. In HTML5 this is encouraged with
<section> and friends when it's semantically appropriate. Don't stress about trying to balance SEO with semantics on this one.
This whole answer assumes that the content of your site is worthwhile. And that you're using all of the HTML5 elements properly.