They both give natural colors, sit in my opinion just as well for a whole day and are easy to work with. If I am to compare really carefully, I probably think that Yves Rocher gives a slightly more natural shade, but of course it depends on which shade you buy.
The big difference is the price! You get Yves Rocher for SEK 63, while Anastasia costs SEK 220.
In the picture I have the shade 02 cendré on Yves Rocher, and the shade Blonde at Anastasia Beverly Hills.
Just to choose by wallet! :)