Test email is being sent fine while order confirmation email is not. How can I troubleshoot?

Check other SMTP or third-party extension 

The issue, when you can send Magento a test email correctly, but other emails get errors, usually occur if any other SMTP or third-party extension is installed in your instance.

It may provoke a conflict which causes rewriting classes responsible for email sending and logging.

Please, first of all, check whether the Amasty SMTP module has conflicts with other extensions by going to System > Configuration > Amasty Extensions > Extensions and Notifications > Troubleshooter > Show: open Models section.

If there are any, you will see the notification in red. In order to check whether other modules affect Amasty SMTP extension functionality, please disable them temporarily via .xml configuration file.

For that, follow the guide below:

•  Edit the following configuration file: app/etc/modules/Extension_name_X.xml (where X is the name or the module you want to disable).
•  Change "true" to "false"
•  Clear Magento cache

Once done, the extension will not affect your Magento anyhow.

The second advise to check is Magento cron.

Our SMTP extension relies on cron performance. In order to send order confirmation emails & invoices it is required to set up cron.

Here you can learn more of about Magento cron and how it should be configured.