If you want to see the Barrier Reef, Cairns is the go compared with Brisbane. There is plenty for tourists to do in Cairns for 4–5 days (especially with a car): trips to the Reef of all sorts, of course, the Kuranda railway, drives north to Port Douglas and the Daintree, the Atherton Tablelands... Cairns itself it not overwhelmingly exciting as a city, being largely a touristic centre, but that's not why you're there.

Brisbane (now Australia's third city) is, as a place, more interesting than Cairns, of course, but it's hundreds of miles south of the Reef. Having spent time in Sydney, you would be far better advised, I think, to go to Cairns than Brisbane.