Hey i am at the University of New south wales ... feel free to ask me any questions
Melbourne is more prestigious than UNSW but UNSW is definitely high on the list of Australian uni's. I have been to Copenhagen and it is a very pretty place. I would say that your main concern is the type of lifestyle you are looking to lead while away. UNSW is situated between the beach and the city. Most exchange students live near the beach (bondi or coogee) and it is a very modern university with 40 000+ students, always things going on. While i dont know much about Copenhagen as far as a place to study i imagine it would be quite different to Australia. Danish are good at english so you dont need to worry too much about getting around town.
oh i forgot to ask what you are studying? If it is business then UNSW may even be more prestigious than Melbourne. There may be an argument there heh. Remember they are in different sides of the world, if you are looking for a name then it depends where you will want to use that name.
I have lived in a few different places and i think there is a foreign experience even if it is a english speaking country. There may not be as much of a culture shock, yet there usually are differences to your home environment.