Yeah!!!
Finally I found the error. First I thought it was the cable, since the interface started disconnecting every now and again and changing sample rates was a pain.
I bought another cable, connected to another machine.. nothing!
So I got another Firewire card with a Texas Instruments chipset instead of the VIA chip that was recommended to me originally.
BUT it wasn't until I found
an article which said it even has to be a
special Texas Instruments chip, namely the XIO2213B.
Now I got a
new card from Amazon and everything works like before!
I still don't know why it stopped working properly out of the sudden. Maybe it had to do with a Windows update. I am on Win 7 x64 here, but it should also work with Win 10.