Company of Heroes clearly isn't close to being fully optimised for its DX10 path as we see huge performance drops right across the board. If anything nVidia is hit harder by the change but both companies clearly need to do some work on optimising this title. That said, frame rates are better overall and it's only the - now established as pretty disappointing - midrange cards that really struggle with this game.
It would seem the way DX10 handles anti-aliasing is more efficient than that of DX9 as it has much less impact on performance when turned on. However it is difficult to conclude at this time as performance in DX10 without AA is so much worse than DX9. It's also worth reiterating, these tests were performed with transparency anti-aliasing enabled and it does have a huge impact on performance due to all the foliage in this game.