Be Number one on Google and Bing:
Search engines constantly update their ranking methods, such as the above speed factor, so keeping up to date and updating your website is important to ensure your rankings are maintained. Converting your site to HTML5 is a forward-thinking way to increase page rankings with search engines by demonstrating that you take your content seriously. Search engines love and thrive on the dedication to rich, well-formed content. Structuring your content with these tags makes it easier to find in Web searches and more relevant to what your users are seeking.  Your content should be the place where your target audience visits.

<address> tag

Facebook:
global time spent on social media sites is up 82% year over year. [ LINK ]
Facebook is even encouraging developers to utilize HTML5 in building the future of the Web. [ LINK ]
With the Social web – fresh content brings relevancy and credibility to your site, which in turn leads to more users. Who doesn’t want returning users, especially those that are directly engaged with your brand?
Social Networking has exploded. More and more people are online and the internet their primary source of information and reference. Your website needs to get on the bandwagon or be left behind. Updating your website to integrate with social networking is considered quite an important marketing strategy.

Have a Super-fast website :
Google now includes page load speed time ads a ranking factor so it’s important your website loads as fast as possible. With new technologies and methods to speed up websites it’s likely your older website could do with a speed boost and gain some extra points with Google
HTML5 include: image scaling, rotation upon mouse hovers, notifying users about errors in real time and assisting local customers to find your business via geolocation. People already expect the Web to be energizing and interactive – now is the time to enhance your site with flourishes that didn’t exist just a few years ago.
An added syntax which allows search engines to better understand the webpage content by dividing information into separate distinguishable parts (i.e. header, footer, main, sidebar, etc.). This division allows search engines to survey a webpage, skipping over sections to better analyze content.
Speed. Quick loading. Low weight.

It Looks great:
Your website may be putting people off using your services or buying your products. It may not represent quality any more. So definitely time for a design update. I would say a redesign is required once every two years.
HTML5 take advantage of the latest techniques and technologies like sliders, image galeries, etc.
we all want a more dynamic website that responds to the user and allows the user to enjoy/interact your content instead of just look at it.

Accessibility:
Mobile browsers have fully adopted HTML5. Allow screen readers to easily access content. Video and Audio Support. Legacy/Cross Browser Support