A common benefit of a private registration is to avoid spam in the form of phone calls, e-mail, and physical mail. Whois information can also be used for different scam purposes. Some people would also like to avoid that they can be connected to different domains. E.g., a person having a business site at domainA might not be interested in making it public that the person also runs a non-related site, blog, etc. at domainB.
For business purposes, the decision between using a private or open registration is a bit more tricky. Public whois information is often seen as a sign of trust and transparency.