The two main benefits of exhaust fans are fume and humidity removal.
The obvious reason we need exhaust fans installed in our bathroom is to extract odors not only from cleaning products that could cause health issues but to avoid unpleasant odors from other humans and animals. Yes that's right, I said animals, I see plenty of litter boxes in bathrooms and they can be pretty pungent at times.
Second, in my opinion as a North Florida home inspector, who is always looking for ways to preserve the life, quality, and safety aspects of a home, exhaust fans are one of the best devices you can have in a bathroom when it comes to removing humidity and odor. When someone takes a hot shower they can create a lot of steam which rises up and settles on the walls and ceiling. Excessive moisture on the wall surfaces will accelerate paint chipping and peeling, it will cause fixtures to rust and wallpaper to peel. The humidity will also possibly create the perfect habitat for mold and mildew growth and if that happens, it's not so easy to remove it completely.
My advice is if you have an exhaust fan use it, if you don’t have one check with a contractor to see if your bathroom is a good candidate for an exhaust fan, if so make sure to have one installed by a professional and make sure it's vented to the outside of your home and not to the attic. By simply having and using your exhaust fan you will be helping your home and your health. A Win-Win!