Lycomings are faster, with a better rate of climb and cheaper to maintain, but cost more per hour in fuel.
Thielert engines have to be replaced rather than overhauled at end of life. They also have expensive components which have to be inspected and overhauled or replaced every 300 or 600 hours.
Thielerts were notoriously unreliable, but the causes have been dealt with, so should now be on a par with the Lycomings for reliability. Thielert/Centurion have recently been bought out by a Chinese company, so are unlikely to go bust again.
Turbocharging allows the Thielerts to operate at higher altitudes, but lack of deicing and built-in oxygen reduces the opportunities to take advantage of that.
The G1000 software in the Thielerts hasn't been updated for 6 years, whereas Lycomings are more up to date.
The Austros are 50kg heavier, which increases stall speeds by 11 kts.
This from someone who has a DA40TDI.