The problem is most likely that you need to authenticate to send to an external email address. Your mail server, most likely, does not require authentication for email addresses with the same domain, which is a common setting.
If there is any type of logging capability, I'm pretty certain you would see an SMTP error 550-554.
Check the program documentation or with the program manufacturer for a way to set authentication parameters.
I see this type of thing all of the time and I am almost 100% certain that this is the cause.