There is no clear-cut distinction between the symptoms of the two - it's more an "if you hear hooves, think horse not zebra" thing. The New York Times article in the grandparent post literally points to the fact that Tom Hanks could get tested for coronavirus based on symtoms that were just like a cold as proof it's much easier to get tested in Australia than the US.

Except that if you read the ABC article, Australia only tests people who both have symptoms consistent with the disease AND either have travelled to an affected country or had contact with a confirmed case. Also, the US press has managed to convince a large (or at least vocal) chunk of the American public that if they can't get a Covid-19 test based on their cold-like symptoms it's somehow proof their healthcare system is falling behind the rest of the world in testing as part of a Trump cover-up, and it's leaking to the rest of the English speaking world. It's all over social media.