Why did the UK, Australia, and New Zealand abolish Brigadier General, and replace it with a Brigadier? But why did solely Canada preserve Brigadier General?
TL;DR: The Canadians did not retain that rank, but re-created it in 1968.
The difference between use of "Brigadier" and "Brigadier-General" in armies seems to depend on the way they feel that senior command should be organised. A "Brigadier" is a senior variety of colonel, while a "Brigadier-General" is the most junior kind of general.
The British Army had Brigadier-Generals until 1922, when they decided to rename the rank "Colonel-Commandant" making it clearly a senior colonel. The title was not well-liked, and was changed to Brigadier in 1928. The armies of the Dominions of the British Empire followed suit, including Canada.
The modern Canadian Brigadier-Generals appeared in 1968, when the Canadian Army, Navy and Air Force were merged into the Canadian Forces. At this point, a lot of traditional practices were discarded and new systems were created, mostly designed by politicians rather than service commanders.
Since then, the Canadian Forces have often seemed a bit strange to other Commonwealth countries. For years, for example, the Navy and Air Force used Army-style ranks, even though members of the Forces could not be considered interchangeable between air, land and sea units. Many of the details of the unification have been reversed over the years, including re-introducing the naval rank system, and separate uniforms and service names for the air and sea components, although the unified command arrangements have persisted. Brigadier-Generals have been retained, possibly for similarity to American ranks.