Your website won't be mobile friendly
Today, many websites are designed to be mobile friendly: visitors using a tablet or smartphone see a site suited to the size and capabilities of their device.
The technology used to build mobile friendly websites (known as responsive websites) is a recent phenomenon and wasn't available when your site was designed.
Your old website won't be mobile friendly and you might deter mobile users who favour your competitor's more modern, mobile-friendly site.
Social media links
The world is immersed in social media. Many modern websites include "sharing" links that let visitors promote the site on Facebook and Twitter. This is simply word-of-mouth marketing in a digital age. If your old website doesn't include social media links, you're not making it easy for visitors to promote your business to their friends and you might miss out on potential customers.
Website styles go in and out of fashion. Today, your old site's design might appear dated.
When a new potential customer arrives on your site, you don't want them thinking negatively about your business because they think negatively about your website. You only get one chance to make a first impression.
For smaller businesses, your website is your chance to compete on a level footing against bigger competitors - don't waste the opportunity!
Don't disappoint your customers
People's website expectations increase all the time. Functionality that would have once been considered cutting edge quickly becomes commonplace. If your site was built years ago and hasn't been improved since, perhaps you're no longer meeting your visitors' needs... and no-one likes a disappointment!