I'm not sure about the log files, but you should have a look at the AD setup for the shared mailbox. I believe this happened to me, that there was a flag within the account setup for the shared mailbox or the underlying active directory that specified whether external access (i.e. Tableau Server) was allowed.
That was my first thought. I will have to look into it. I would have expected a Powershell Send-MailMessage command to fail, using the same parameters from the Tableau server, but that all works fine.