As another Tasweigen (albeit of mainland birth... no second head or amorous encounters with siblings), let me through a few thoughts out:
First, what is the POD? If the French attempt to colonize in the 1800-1815 timeframe, and events further afield are generally inline with OTL, either the poms roll in and boot 'em out or the colony's early development is badly stifled... Putting things simply, the early prosperity of Tas was based on three things: Agricultural exports principally to NSW (the Sorell district near Hobart was once the breadbasket of the colonies); whaling and sealing. With the Napoleonic raging number one is out big time, number two is much more limited than OTL (there's a blockade limiting how much you can reliably ship home, thus implying vastly reduced profits relative to OTL), number three was largely conducted outside areas of real control of any government. Additionally, the blockade will make shipping colonists out from France difficult (relative to the poms sending people to Aus IOTL), so I suspect a more likely source of migrants would be the French Indian Ocean colonies of Mauritius and Reunion (but IOTL both taken by the poms 1808-1810...), with interest results in terms of demographics (French, Indians, Africans, Chinese... oh my...).
Of cause, you could forestall that by having the Napoleonic Wars end earlier... Also could generate some interesting butterflies if the poms don't boot 'em out (earlier colonization of OTL Melbourne or more rapid development of the Hunter Valley as alternate food sources for NSW?).
The other alternative is the French explorers show less interest and the poms don't get startled into acting in 1803/4. Once things back in Europe calm down 1810-15ish then the frogs go a colonizing...
Second, longer term? Broadly speaking, Tas's economics up till circa 1900 can be summarized as: Grain and whaling 1805-1840ish; gold rush 1850ish, overshadowed by Victorian equivalent (Melbourne rises from a handful of mud huts to one of the leading cities in the Southern Hemisphere, putting Hobart in the shade); back to agriculture (more sheep and wool than grain); tin rush 1870s; development of west coast lead, zinc and copper mines (1880s on). The gold and tin rushes have a significant chance of seeing largescale migration, including probably a fair few Anglo-Australians and possibly Chinese (per OTL in Victoria during the 1850s or Tas during the 1870s tin rush).
Aborigines? This depends... the French were rather variable in their treatment of native peoples, but then again, so were the poms. I suspect it comes down to early development and growth of the colony... if (per scenario one) growth is relatively slow then I see assimilation as more likely then OTL's near extermination. Still, either way, come 1900ish the Tas Aborigine population will be a small minority...