Ask to Move In - adds the sim to your household. They're just like any other sim in your household, controllable by you. This interaction usually only appears after some friendly social interaction (in the top left corner it will say "simA thinks simB is being sociable/friendly.")
Ask to be Roommate - does not
add the sim to your household, but does make them live with your household. The sim will live in your house similar to the way a butler does. They will keep their job, they will leave the house, they will eat, sleep, watch TV, bathe, etc, but they can't be controlled by you. Once a week they will pay rent. The roommate does
remove a household slot from the pool, so keep that in mind. For example, if you had two playable sims and two roommates, the game counts that as four sim slots being taken up out of the possible eight maximum. You would only be able to add four more playable sims, four more roommates, or any combination that adds up to four to the household.
So for sims you don't have to control, you'd be wanting to add roommates. The game will do the controlling of the roommate for you.
Hopefully that helps sort it out!