If they did that, the original company could lose those customers, and the employee in question would likely have proprietary information, such as how much their first company charged that client.
Additionally, non-competes are common when a business is sold. If you bought a business, whether it's a small business or something larger, you would not want the seller to turn around and start a competing business, undermining the value of the business you purchased.
Finally, protecting confidential information is another instance where a business might use a non-compete clause to protect such data. For instance, if you knew the secret formula for Coca-Cola (KO -0.29%), you would not be able to share it with a competitor.