I say this not only as someone having lived in both, but as someone who has had some past professional interest tracking rents and the like: Sydney has by far been the more expensive between the two cities in terms of rental and house prices.

There may be a subjective element in being present in Melbourne while rent prices took off which makes Melbourne seem to have become more expensive, but it's not even close.

Edit: for instance, I live within walking distance of Melbourne cbd in a modern 2 bedder with 1 car space. Market rent approx $650 - $750 pw (not even including aberrations like docklands). In Sydney, you have to draw a substantially wider radius and transport time around the city centre and employment hub before you start hitting comparable properties and rates.