What does a differential do?
A differential typically serves two functions: redirect power flow to an axis in line with the drive wheels, and to allow a speed differential or difference between two drive axles to ease turning the vehicle. The amount of speed differential allowed between sides varies with the design of the differential used. The video on the right is very old, but is an excellent explanation of what a differential is and why it is necessary for your car (skip to 1:49 if you want it to get to the point).
A transfer case is a part of a four wheel drive system found in four wheel drive and all wheel drive vehicles. The transfer case is connected to the transmission and also to the front and rear axles by means of drive shafts. The transfer case receives power from the transmission and sends it to both the front and rear axles. This can be done with a set of gears, but the majority of transfer cases manufactured today are chain driven. On some vehicles, such as four wheel drive trucks or vehicles intended for off-road use, this feature is controlled by the driver. The driver can put the transfer case into either “two wheel drive” or “four wheel drive” mode.