From my experience Sydney is a more vibrant City and more walkable for a few reasons
It has MUCH more density of population, especially around the inner parts. This makes it feel like the city keeps going, even when you leave the CBD. Melbourne seems to go from CBD to medium/low density very quickly. The density also makes things closer together and therefore more walkable.
Population density: is Sydney an outlier? | The Urbanist
The public transport is probably about the same, however quite a few more people use public transport in Sydney than Melbourne which means better services, for example trains on the weekend start earlier and finish later in Sydney. From my experience Sydney's train network is more advanced. More people use Sydney's buses than Melbourne trams. The gap is about to get wider with Sydney starting to introduce proper metro, driverless trains.
How do people get to work in Sydney and Melbourne? - SphereSphere
Anytime Sydney is mentioned in a forum you will get 100's of people from Melbourne pounce and say how Melbourne is so much better, mostly by sweeping statements that are not quantifiable. It really seems that the tide has turned, Sydney is now the underrated one that actually has a lot of offer. Forgetting the tourist trap areas (Bridge, O House) where locals really don't go near, Sydney is quite an amazing place.
Oh and by the way I LOVE Melbourne too (going there over easter) and really think you can't loose but when addressing your criteria I think Sydney would win.