Lets not get things mixed up here:
For two heatsinks of equal size, one made out of copper will have a much larger thermal capacity. This is mostly explained by coppers greater density: its heavier, so there is more material/mass to heat up.
This does not matter for continous operation though: the only thing that matters there is thermal resistance, and that varies mostly with size and surface area, and not that much with thermal conductivity of the metal used.
If you made two heatsinks for equal -weight- instead of size, the one made from aluminium would be superior in almost any application imaginable. This is also the reason power lines are often made from aluminium: its conductivity is lower than that of copper, but using thicker wires makes sense since their mass (and cost) remains lower than the copper equivalent.