"Australia Day" is January 26th. Commonwealth Day is May 24th. (It used to be Empire Day). I doubt anyone bothers with it at all. May 24th was the birthday of Queen Victoria. The day is still a holiday in Canada I think.
More people acknowledge Australia day this century than they did in my youth. The problem with the date is that it is already in School summer vacation so is nothing special - and Australians are really not patriotic (except in following sport).
Australia Day is the anniversary of the invasion of this continent by Europeans.