Thanks everyone for the replies. So if I took the Bachelor of Advanced Computer Science (Honours) at UNSW, and drop Honours, then it will say Bachelor of Advanced Computer Science on my testamur? Because there is only an honours degree for this specific (advanced) course at UNSW, not a normal 3-year one. (There is also a Bachelor of Computer Science (Honours) degree, which I assume is the less intense version of the same degree. There is no just 'Bachelor of Computer Science' degree).
Also, would having an 'Advanced' degree influence an employer's decision to hire me, more than a normal degree would?