One of the biggest advantages of using HTML5 now is that, as the next standard for web development, it provides a certain amount of future-proofing. Previous HTML specifications have allowed a certain amount of interpretation, which has resulted in inconsistencies between browsers; the results of which range from minor display issues to serious impact in functionality. The HTML5 specification seeks to remove these inconsistencies by having clear instruction on how browsers should handle pages, which should result in a better experience for users, and easier development and testing for programmers.
Sites built using HTML5 should be more stable, both on today's browsers and future versions, and are increasingly supported by mobile phones and tablets, thus removing the need for building alternative websites using older mobile-specific technologies.
As HTML5 is based largely on HTML4, it's also backwards compatible with older browsers - although a notable exception is even quite recent versions of Microsoft's Internet Explorer (a workaround for which is freely available, bringing harmony to the web). Following the HTML5 best practices of semantic markup and separation of styling, HTML5 pages can easily support text-only and specialist browsers such as screen readers and braille devices, greatly improving accessibility.
Traditionally, the approach taken by site builders was "graceful degradation" - sites were built to always display their best features, but in the event of browser incompatibility, to fail gracefully and display what they could.
Although the HTML5 (and the corresponding CSS3) standard hasn't been ratified yet, web browsers are increasingly supporting the new features based on the current revisions. The core parts of HTML5 are considered pretty stable now, and are only waiting for the final "rubber stamping" - any edge cases discovered now will be measured against the current implementation to define the behaviour without breaking existing sites.
As a result, it is generally considered to be safe to use these new technologies, based on the assumption that there will be few changes made before the final ratification of the standards, and that any changes will (if possible) be minor or easily handled.