They way I differentiate two, is TSM (Tableau Services Manager) is the backend admin account and is used for configuring and licensing Tableau Server.
The Tableau Server Administrator account is used to administer the content published to Tableau Server, user/ groups and permissions.
TSM is automatically installed when installing Tableau Server, and the account used to log into TSM is typically the account being used to remote into the machine hosting Tableau Server. I found a decent guide using their online help.
https://onlinehelp.tableau.com/current/server/en-us/sign_in_tsm.htm
Also, a good way to verify which user can/is being used for TSM is to try running a TSM command from CLI. For example, running ' TSM Status -v ' should prompt you for a password for X user, this is what you would log into the TSM UI with.