The information populated in the Exchange Properties section is retrieved by Adaxes directly from your Exchange server using Exchange management cmdlets. In particular, the display name is returned by the Get-Mailbox cmdlet. If you change somthing inside the section, the information will be also sent to the Exchange Server using Exchange management cmdlets, for example, Set-Mailbox.
The display name parameter in the Exchange Properties section is the Simple Display Name returned by Exchange management cmdlets. This property can be set for a mailbox using the Set-Mailbox cmdlet using the SimpleDisplayName parameter. The attribute that is returned / changed by the cmdlets is always Simple Display Name.
As far as we know, Exchange Management Console also shows the Simple Display Name, however Exchange Control Panel available in Exchange 2013 shows the Display Name attribute instead of Simple Display Name. The cmdlets return the Simple Display Name, however. Hence the difference.