If you stay in the bath long enough, your whole party gets the Toasty status for a while, which gradually heals them each turn in battles. (Forget if it heals on the world map, too, or not.) There are also some special scenes you can unlock by bringing different groups of people to the bath (these scenes also depend on which version of the bath you enter). You can also decorate your bath with ornaments, dolls, statues, and paintings, and certain combinations of these will unlock additional scenes/decorations.
More or less, it's just an extra, with not much significance, but if you want want a free Toasty status on your whole party for a while, or just want to see these extra scenes, this is where to go.
For more details, go
here.That should cover just about everything there is to do with that bath.