The only property that is of relevance to heatsink design is 'thermal conductivity' - copper is the winner here (discounting silver due to its cost).
The thermal capacity/diffusivity, etc. are irrelevant.
While Aluminium will store more heat energy in it, all this means is that with copper, the heatsink will get up to running temperature faster when the CPU is switched on, and cool down quicker when switched off.
Some people say that aluminium 'holds onto heat longer' as if this is something bad for heatsinks - it isn't. It's irrelevant. All it means is that the heatsink temperature changes more slowly - it has no effect on the actual temperature it gets to.
Of course, heat pipes change things - heat pipes have 'conductivity' tens, or even hundreds, of times higher than copper. This allows heat to be moved a long way to multiple fins.