I live in Brisbane, and I visited London for a month very recently. Australia is more expensive; it has a higher cost of living. There will be select items that are cheaper in London, but the general expense of living in Australia is far higher. This is reflected in Australia's minimum wage, which is almost double that of London's (assuming you are 21+ in Australia). It also really depends on your interests, but I think Australia is culturally premature in regards to its music / nightlife / youth pop. In Brisbane, we have one 'indie' radio station, Triple J, which is essentially attempting to monopolise the music market with their idea of a trendy sound. If you are an up and coming artist, you need to get your shit featured on Triple J to have almost any hope of success.
As I'm sure other people have mentioned food and public transport in Australia isn't great. The food is ridiculously expensive, and you would be hard-pressed to get anything decent for under 10$. Trains and buses are acceptable, at best, they're relatively infrequent and unreliable compared to London (or Shanghai, where I have also lived). Going out and drinking here also sucks ass, because of aforementioned shitty music and homogenous nightclub culture, also drinks are pricey when not bought from a liquor store. Very pricey. You also can't buy liquor here in convenience stores, only designated liquor stores. Everything also closes at around 7-8, with the exception of supermarkets and convenience stores, which close, at the latest, midnight. This Brisbane specific I believe, but walking around here is also kind of annoying because everything is on a ridiculous incline and there's often stretches of nothing but residential area outside of the city.
tl;dr I would live in London over Brisbane if I had the choice.