It looks like I am talking only to myself. But there where quite a few onlookers to my thread. So I am writing a conclusion to my problem.
Here are my test results:
With new driver (AMD_Radeon_HD_7790_126.96.36.199-130416a) I still got some wild hair in the integrated Benchmark. I could still make the first picture above. But than the hair became smoother. I think in the whole benchmark turn the hair was ok in 80-90% of the time. In the game the error did not occur or I haven't seen it.
Than I installed Phenom 9550 Quad Core instead of Athlon II 4200+ and installed 4GB DDR2 800 instead of 3GB DDR2 533 which I got from a friend.
After that I made the tests again. And what should I say: No errors whatsoever.
I think most likely the memory made the change. The framerates are mostly the same as before.
Look like you really should have at least 4GB Memory for TressFx.