It's just another set of full-size rooms to decorate. You can go for themes, if you want to, and show them off to visitors in your town, but there's no explicit in-game reward for making an exhibit. They're more for personal satisfaction.
Like, just for example, I'm slowly working on getting miniature models of all of my already-completed fossils, and plan to make an exhibit themed around them, eventually. Right now all I'm really doing with them is using Exhibit 1 as a gyroid dumping ground, although in and of itself that's actually grown to be pretty entertaining. The room is just packed with those freaky little guys.
Is this finally a conundrum that CAN'T be solved by helicopter theft?
I'm using two exhibit rooms, one for the Balloon series and the other for all my Fortune Cookie items. The Balloon series just because it's ugly but I still feel an urge to collect all the pieces without wasting storage/wasting room in my house and while I like some of the Fortune Cookie items I don't want to dedicate a room in my house for them.