When using an external SMTP client such as Outlook to send an email, the sent mail is not saved into the account's "Sent" folder. They will not show up when looking at Webmail or through a different device.
When using Webmail to send an email, they are automatically saved to the "Sent" folder of an email account. However, external SMTP client will not have this configured by default, and messages will not be automatically copied unless the client is configured to do so.
You can configure Outlook to save mail to the Sent folder with the following steps:
In Outlook 2010 or later:
1. Select "File » Options"
2. In the resulting dialogue, select "Mail"
3. Under the "Save Messages" section, enable the option to "Save copies of messages in the Sent Items folder"
4. Select "Ok"
Messages now sent from Outlook will be copied into the server's Sent Items folder.