People often talk about docking and berthing. They both seem to be connecting a spacecraft to the ISS, yet seem to be very different. Why?
The terms docking and berthing have a nautical origin. Smaller ships come into port under their own authority and dock. Large ships instead are berthed. They come to a stop outside of the port, relinquish control to the port authority, and are towed into port by tug boats.
Docking with the International Space Station is essentially a controlled collision with the station. This controlled collision has to be extremely precise in order to be nondestructive. Because the space station is an extremely expensive resource, approving a vehicle that plans to dock with the station is something done with extreme care (and extreme expense).
Berthing opened up the door to automated operations by the Japanese HTV, SpaceX's Dragon, and Orbital's Cygnus. With berthing, the designers of a vehicle need to prove to NASA that their vehicle can safely approach the station, that it can hold position and attitude in well-designed berthing box, that it recognizes and responds to problems along the way, and that it relinquishes control to the ISS upon command. That the ports used for berthing are larger than the ports used for docking is a side benefit.
The basic difference between docking and berthing methods is as described by @OrganicMarble and the linked article, that berthing involves the robot arm and docking does not.
The reasons why one is used rather than the other was discussed by the NASA PRO in a recent webcast. Obviously, if the vehicle is not capable of docking it must be berthed; however it must be equipped with a berthing connector and a robot arm connector. These things must then be designed and built-in.
One of the reasons why one is selected in the design rather than the other is cargo size. The berthing and docking ports on the ISS are of different shapes and dimension and can thus be used to transfer different volumes of cargo. In particular the Common Berthing Mechanism (CBM) used by the H-II Transfer Vehicle (HTV) and the Dragon spacecraft have a square opening of 1.3m (50") which can transfer whole equipment racks. By contrast the Russian and new US docking port is a circular port of 0.8m (31"), which is too small for racks.