Drum brakes work by applying friction from a pad inside an enclosed drum. The drum is part of the hub of a wheel. These type of brakes generate a great deal of heat and warning labels appear on the outside of the hubs warning the rider not to touch the hub for a time after the brakes have been applied.
Disc brakes work very much like caliper brakes, with a separate disc attached to the hub.
The main benefit is that the disc is away from the wheel spray and consequently any liquid, dirt or other materials. Both of these types of braking systems add more weight to the bicycle, but drum brakes are especially heavy. These types do appear on professional downhill bikes where the added weight is not a concern and the added braking power is essential. Downhill riders like Missy generally ride heavier dual-suspension bikes. Since the journey is downhill only, weight is not as much of a consideration.