Copper by far is better at conducting heat and better at dissipating through convection. Why? It doesn’t matter what the material is made of, what matter is the temperature difference between ambient and delta. The bigger the difference the faster the heat transfer, ergo copper (since its better at conducting heat) will be better at dissipating it since its able to get itself “hotter”.
I hope that made sense…
Source: Smith, Johnathon, 2005
As a kind of counter situation which some may be trying to prod at though, the specific heat of copper is .0923 cal/gK and it has a density of 8.92g/mL. Aluminum has a specific heat of .215 cal/gK and a density of 2.702 g/mL (Chemfinder.com).
If you take those and figure out the heat capacity per volume, then you get 0.82 cal/mL*K copper and 0.58 cal/mL*K for aluminum. It seems like some unnamed computer sites may think that their equivalent copper heat sink takes longer to cool down than the aluminum one because per volume, the copper will store more thermal energy per temperature. That is very true, but it does not deal with how well the metals conduct heat or anything else that is often implied.