There are two email addresses that need to be checked here.

Firstly, the desitination email address which is presumably the same for both your store customer and your SMTP test.

Secondly, the from address.  You configure an email in the store admin settings, this email is used as the from address for all emails from the store.  If your SMTP server is not set to allow relaying from this address then your email will not send.  If possible, use the same from address as your SMTP test, or else check your SMTP server logs for declined requests (if you have access).

If you have verified that both your to and from addresses are identical in your store setup and in your SMTP test, then there is a problem other than configuration here.