The word sections comes to mind first.
You'd often hear about a VIP section at a fancy night club, which is an area for special guests that not just anybody is allowed to enter. It might be up on a balcony, or divided by a velvet rope. And other parts of the club could similarly be called sections as well.
However, depending on the club's layout, rooms might be more appropriate. "Rooms" denotes that the sections are mostly or completely divided by walls. And if the VIP section was actually closed off by walls and a door or curtain, it might be more properly described as a VIP room.
For use in your example scenario, the different sections of the Moon nightclub might each have their own atmosphere or vibe.
If a club's music and atmosphere are pretty much the same throughout, you may just refer to the specific parts of the club instead of giving a general name for the parts. For example, a club might have a dancefloor, a bar, and a seating area or tables.