Sydney's rental market is noticeably more expensive, especially if you're aiming for an inner-city location. Expect to pay a premium compared to Melbourne for similar apartments. Groceries are fairly comparable, though you might find slightly cheaper options at markets in Melbourne. Public transportation is generally good in both cities, but factor in Opal card costs in Sydney and Myki in Melbourne. Childcare is expensive across the board, but anecdotal evidence suggests Sydney is, again, a bit pricier. Hidden costs? Sydney's toll roads can add up if you're commuting. Melbourne has more affordable dining out options, IMO.
GlobalCareerGuy: Don't forget that Sydney's higher salaries can offset some of those living expenses, depending on your field. Melbourne's culture scene is great, but factor in weather; Sydney wins on sunshine hours.
budgetliving: Sydney's rental prices are the real killer; budget accordingly if you're prioritizing location. Melbourne definitely wins on the dining scene.