You may find that anonymous relay to external email addresses is not working in Exchange 2013. To enable anonymous relay in Exchange 2013, you need to run this PowerShell command to make it work:
Get-ReceiveConnector “” | Add-ADPermission -User ‘NT AUTHORITY\Anonymous Logon’ -ExtendedRights MS-Exch-SMTP-Accept-Any-Recipient
I guess Microsoft implemented this so they could protect administrators that didn’t know what they were doing, from accidentally enabling anonymous relays to external addresses, because obviously you only enable anonymous relay if you absolutely have to for security reasons.