This almost comes under 'opinion based', but we can treat it as a compare-and-contrast option. Hi from Sydney, BTW.
Australia is huge. Literally similar in size to the US. That doesn't mean you can't drive across it, but you have to take things into account that don't occur to people compared with the US:
- gas stations are few and far between in some places
- there's long distances with nothing in-between
- if you break down, you might not see anyone else for hours, or days.
There's a reason the main tourist trail is on the east coast - it's where most people live, and where most of the interesting stuff is in terms of variety.
A common backpacker route (if doing a circle route) is to go from say, Melbourne to Sydney (perhaps via Canberra), up the coast to Byron Bay, Gold Coast and the Sunshine Coast, Cairns etc, then perhaps inland/across to Darwin, down to Alice Springs (Uluru, woo!), through to Coober Pedy and Adelaide (with Kangaroo Island), and back to Melbourne, or at least a similar route with alternative starting points (perhaps Sydney).
Perth is usually considered a flight - most wouldn't consider driving, busing or training - although all are options. I've bussed from Adelaide to Alice Springs, and that alone is 20 hours.
I've had relatives who lived in Derby (near Broome) for a while, and there's not much else around. I'd love to do the drive from Perth to Broome, but it's long and there's not much in-between to look at.
So, it depends what you're after. If you've got the time to do a trip that you probably won't get much chance to head back and do again, maybe consider the harder Perth to Broome. If you're after seeing the sights and places that Australia is known for, then you're better off on the east coast.
If you're going to do both, fly or bus. You don't want to be the tired driver driving in twilight, dodging kangaroos and worrying about the gas light that's been on for 50km. Unless of course, that's your thing ;)