Actually, Translink is the transportation authority for Greater Vancouver, which includes the transit system but also includes responsibility for roads and highways and AirCare. BC Transit is a separate entity responsible for public transit in BC outside of Vancouver. It used to include the Vancouver Regional Transit System until the formation of the Greater Vancouver Transportation Authority, GVTA, in 1999. In early 2000 the GVTA was renamed Translink. The various components of the transit system in what used to be the VRTS are now under the umbrella name of Coast Mountain Bus Company, which was chosen after a naming contest.
(Allah alone knows how that horrible name won the contest. The new name is supposedly to increase "brand recognition", ignoring perhaps that there is a retail chain in Vancouver called Coast Mountain already. Besides, changing the name has only increased confusion. It's not like a monopoly needs brand recognition anyhow.)