Here is a real world example
Go to
https://www.bankofamerica.com/
Everything looks fine. The CN is
www.bankofamerica.com, the SSL cert is valid for today, and it is singed by VeriSign, who my Browsers trusts. All 3 are good, no errors
Now to go
https://bankofamerica.com/
Now I get an error, same website, same cert. But the CN does not match. In Firefox I get this error:
bankofamerica.com uses an invalid security certificate.
The certificate is only valid for
www.bankofamerica.com
Date is ok, and singed by is ok, but 1 item is not, the CN is no good, so error we get.
So if they told you that
www.example.com would work for non-www, well they are right in regards to the Encryption still works, but the authentication/trust will not, the browser will complain, and in today's world of phishing and stuff if your customers see this they will flee!