They're the same. The difference depends on the following words. For example: "I've just moved into my new house." Use INTO when you say the place after. "I bought an apartment. I'm moving in next week." Use IN when you do not say the place after.

Thank you, you're so kind!!

Normally you would say “move into” before stating a specific place. For example, “I’m moving into my apartment tomorrow”. “Move in “ is more general and doesn’t need a specific place. “I’m moving in tomorrow.” “Tomorrow is move-in day.”