Diamond isn't better at conducting heat due to the way it bonds on a molecular level, but it's surely a tad more expensive. The reason people say that it's better is due to confusion of (W/mK) which is not a good indication of diamonds ability in this regard.
There are things superior to copper and silver though, synthetic carbons for example. A few companies have utilized this in prototypes but never made it to the manufacturing line due to costs and difficulty in processing involved.
Yeah I wish some of those prototype materials will make it out. They are extremely expensive stuff. These light nanotubes or whatever they are look to b epromising though.