Do not disturb can be set as an action in both the modes. If you want to adjust other settings along with Do no disturb, then modes or better.
Modes allow you to change multiple settings when certain conditions are met.
I turn on Theatre mode when I go for a movie. Have set it to turn on manually and the duration as 2 hours.
Sleep mode has sleep schedule option to turn on the mode automatically + wake up alarm option (which are not available in the Theatre mode). You can ofcourse turn on the modes manually, in which case they are all the same. Sleep mode is integrated with the Clock and Samsung Health apps.