Sounds like you are using IMAP email. Thunderbird will create a Sent folder if that is what is checked in Tools-Account Settings-Copies and Folders. My guess is that the IMAP server create the other Sent Items folder. You could try eliminating the sent choice under server settings and selecting Other and the IMAP server folder.

Sometimes I use Thunderbird, and sometimes I use Outlook for the same email account. Is this the difference?

I don't use Outlook. Maybe they have a support forum for their software.

If anyone was interested I was able to find a relevant question here: https://support.mozilla.org/en-US/questions/1015564

Emails sent from Thunderbird go into the "Sent" folder. Emails from Android phone go into the "Sent Items" folder. MS Outlook sees both, but TB only sees "Sent"

This webpage was helpful: http://helpdesk.princeton.edu/kb/display.plx?ID=9430

Sent vs. Sent Items
In Thunderbird, the Sent folder (located on the IMAP mail server) retains a copy of each of your outgoing e-mail messages. This is a function that can be altered in the preferences, but is most often configured in this way on the Princeton campus. Outlook does not recognize that the Sent folder is any different from any other e-mail folder. Outlook's equivalent of the Sent folder is called Sent Items and is generally stored locally rather than on the central mail server. When you begin using Outlook, copies of your outgoing e-mail messages will be stored in the Sent Items folder. The Help Desk recommends that you create a rule to send copies of your Outgoing messages to your Sent folder on the IMAP server. See Solution 9202 for instruction.

This one also: http://office.microsoft.com/en-001/office-online-help/set-up-email-in-mozilla-thunderbird-8-0-HA102823188.aspx

If you’re using an incoming POP3 server, you can only synchronize your Inbox folder. If you’re using IMAP4, you can choose the folders that you want Thunderbird to synchronize with the server. Remember that IMAP4 doesn’t support the synchronization of the Contacts and Calendar folders. Thunderbird uses a folder named Sent to store sent items. Exchange and Outlook Web App use a folder named Sent Items. You can synchronize the email items that you’ve sent in Thunderbird with the Sent Items folder on the Exchange server. To do this, follow these steps:

This is about the “sent mail” rule: http://www.ucl.ac.uk/isd/staff/mail/configure/thunderbird/sent_mail_rule

I think I would like to delete the sent folder completely and just use the sent items folder.

Will it be possible to use just one folder?