A new 9 pin connector was developed to support 1394b Standard. In addition to higher possible speed, 800 Mbps, it also had a different signaling method. The old mode was then named alpha mode, or "1394a" mode. Devices that were designed to only operate in beta mode were given beta connectors so that alpha devices could not plug in to them. A device that could work in either alpha or beta mode would have a 9 pin bilingual socket that would allow either alpha (1394a) or beta cables to connect.
New devices using 1600 and 3200 Mbps are beta only and feature 9 pin beta sockets.
Most devices would want a bilingual socket. Only if you want to restrict your devices to 1394b only would you use a betasocket.
Beta cables allow the connection between two 1394b 9 pin devices. A bilingual cable allows the connection between 6 pin or 4 pin devices to 9 pin sockets. The beta plug fits into both beta and bilingual sockets. Bilingual plugs can only fit into bilingual sockets.
See the diagram bellow. A bilingual plug will not go into a beta socket,though all other options are possible.