Tanning is the skin's response due to the dose of UV radiation that the person is exposed to [Wikipedia].
One major factor for the differences in tanning between the two hemispheres is that the Earth's orbit is not circular - and is at aphelion (furthest distance from the sun) during the Northern Hemisphere summer and is at perihelion (closest distance from the sun) during the Southern Hemisphere summer. See the diagram below:
Hence, a greater amount of solar radiation reaches the Earth's surface during the southern hemisphere summer.
As for the ozone 'hole', as can be seen in the image below, it only really affects Antarctica and surrounding oceanic waters to any great extent:
Taking Australia as an example (as you mentioned it in your question).
According to the Australian Bureau of Meteorology FAQ about ultraviolet radiation:
Does the Antarctic Ozone Hole ever come over Australia?
No. The ozone hole has only ever been observed to be well south of the Australian mainland and Tasmania. In fact, during springtime, when the hole is in existence, ozone levels over southern Australian cities are at their highest.
However, after the ozone hole has broken up parcels of ozone depleted
air mixed with mid latitude air move northwards. These parcels can
move over the southern part of Australia and cause a reduction in
total ozone values.
Australia's SunSmart FAQ about UV radiation sums it up nicely:
Why does Australia have high levels of UV radiation?
Australia experiences some of the highest levels of UV radiation in
the world because we are close to the equator and have a lot of clear
blue-sky days. The Earth's orbit also takes countries in the southern
hemisphere (Australia included) closer to the sun in their summertime
than countries in the northern hemisphere during their summer.