The short answer, for users of Australian (and British) English is that it generally doesn't. However, the usual caveat applies: if the proper name of something includes a variant spelling, you're stuck with using that version.
One of the more consistent themes here at Mind Your Language is that we shouldn't be adopting American English spellings simply because we run into them constantly online. That's not a criticism of our US brethren who are using those spellings; it's simply a reminder that what's correct for one group of people isn't correct for another, especially in the case of a global language such as English. Accurate writing and spelling requires learning lots of rules, and regional usage is a key element of that.
One rule which I'm reminded of this week is that even when there is a clearly correct spelling in general Australian usage, such as 'centre' rather than 'center', there will be occasions when you have to use the Americanised spelling. I'm in the middle of one such occasion: Data Center World in Las Vegas, which I'm attending for our ongoing World Of Servers series.
I covered an event called Data Centre World for World Of Servers back in February. That one was held in the UK, so there wasn't a spelling issue. But this time around, I'll have to be happy with providing data centre advice from an event called Data Center World. It would be confusing and inaccurate to change the name of the event, which is essentially serving as a proper noun, to match my normal spelling practice. Rules have exceptions, and you need to learn those.
Anyone who codes HTML will already be familiar with this problem. The standard tag for putting text or images in the centre is
center, and any attempt to use the alternative version won't work. That doesn't thrill me, but it does remind me that accuracy often depends on context.
Lifehacker's Mind Your Language column offers bossy advice on improving your writing.