What's the difference between AirBnb and Booking.com?
The difference is that on booking.com you book commercial establishments which main business purpose is to provide accommodation to travelers. On AirBnB you're booking - as Roddy of the Frozen Peas said above - someone's else personal property.
What is means to you:
Services available. AirBnB is similar to as if you stayed in your friend's home. So usually you will not receive typical hotel services such as 24hr front desk (and the ability to check-in at 2am), luggage storage, or laundry services. The check-in and check-out hours are usually limited, and often require advance coordination.
On the other hand you might get extra services, such as host could meet you on a nearby train station, and walk you through.
Reliability. If something is broken during your stay, such as you returned to your room at 11pm and found out the air conditioner doesn't work, or a clogged bathroom - you generally have a much better chance to get it fixed in a hotel. They will have maintenance people on call, and they can just move you to a different room (even in a different hotel if they're full). For AirBnB the host is limited in their abilities, even if he/she is a handyman, and moving you to a different room/apartment is usually not an option.
Flexibility. If you need to extend your stay upon arrival, or arrive early, a hotel usually would be able to accommodate you (often only at rack rate though). With AirBnB it is much harder to achieve.
Lack of guarantee. Realistically your reservation on AirBnB is not guaranteed, and those are routinely canceled because the hosts changed their travel plans, or simply found out a big event is coming and they could re-rent and make more money. Just search Twitter for "airbnb canceled" and you will see a lot of those every day.
There are advantages too, however - if you travel in a large group, you can rent a bigger place than a hotel. You'd stay in a more home-like environment, often in a good/exclusive area, and the price for accommodating that number of people may be cheaper.