The Schedule E is used to report rental real estate income, among a few other types of income. You would use Schedule E if you did not provide substantial services to your guests and had a more passive role in earning Airbnb income. Most Airbnb hosts would likely report their income on a Schedule E.
The Schedule C is used to report business income. In short, you would use Schedule C to report your Airbnb income if you treated your rental property like a business. You'll know that you were running your Airbnb rentals like a business if you were providing "substantial services to the people who rented your property. You can read more about substantial services here (under point #7).
If you provided "substantial services" to your renters, then you're considered to have had an "active" role in earning Airbnb income. That active role indicates you should be classifying your Airbnb activity as a business on a Schedule C.
As you can see, the most important factor in determining whether you file with Schedule C or Schedule E is to know if you were providing substantial services. We would recommend using these lists to determine where your income falls—under a Schedule C or a Schedule E.