There is at least one big advantage that the air outside the house has over the air inside the house: On a sunny day, the air outside the house comes with a good dose of ultraviolet radiation from the sun. While ultraviolet rays are not so great for your skin, it turns out that they are good disinfectants -- they kill bacteria. So if you haul a rug to a sunny spot on the driveway so that it gets several hours of strong sunlight, you are using the sun to disinfect it. With something big like a mattress or an area rug, it's one of the few inexpensive ways to kill bacteria -- you can't exactly throw a mattress in the washer!
It turns out that ultraviolet light gets used as a disinfectant in a number of different products. For example, you can buy swimming pool systems that disinfect the water in the pool by running it past a strong UV light. Some municipalities do the same thing to augment the chlorine that treats drinking water. You can even buy a device that uses UV light to disinfect your toothbrush!
In third-world countries, there have been experiments using clear plastic soda bottles filled with contaminated drinking water. People leave the bottles in the sun, and it actually helps to reduce bacteria in the water. See, for example, Sun and water: an overview of solar water treatment devices.