Copper or Aluminum Heatsink?
Copper has got better thermal conductivity.  

Aluminium - \$ \mathrm{ 200 \frac {W} {m\cdot K} } \$
Copper - \$ \mathrm{ 400 \frac {W} {m\cdot K} } \$
(from here, also here)

But thermal conductivity within the solid material is only a part of the story. The rest of the story depends on where one wants to dump the heat into.  

Liquid coolant

Copper heatsink (one may also call it heat transfer block) will perform better than aluminium.

Air with forced convection

In other words, there's a fan blowing onto the heatsink.  Copper heatsink will perform better than aluminium.

Air with natural convection

I've saved best for last.  It also looks like it's the O.P.'s case too.

With natural convection air, the copper heatsink perform only marginally1 better (in °C/W) than aluminium.  This is because the bottleneck isn't in the transfer withing metal. When you have air with natural convection, the bottleneck is in the transfer between metal and air, and it's the same for Al and Cu.

1 I might add that the marginal increase is often not worth the cost of Cu.


  
  This curve demonstrates the non-linear relationship between heat transfer and material thermal conductivity. The curve is generic.  It applies to any application having both conduction and convection components to the total heat transfer.  [Radiation is typically small and is ignored in this calculation.]  The shape of the curve is the same regardless of the application.  The quantitative values on the axes are not shown  because they depend on the power, part size and convective cooling conditions.  They become fixed for any given application and set of conditions.  It’s obvious from the shape of the curve that heat transfer depends on material thermal conductivity but there is also a point, a knee in the curve, where increasing thermal conductivity produces negligible improvement in the heat transfer.
  (source, emphasis mine N.A.)


Phil have already linked an ECN article which compares aluminium and copper in air with natural convection.  Here's another take at it: what if we compare aluminum to a material with lower thermal conductivity (unlike copper).  There is a company that makes thermally conductive plastic.  It has got \$ \mathrm{20 \frac {W} {m\cdot K} } \$  conductivity, but that's a lot for a plastic.  They have benchmarked it against aluminium in air with natural convection.


E2 is the plastic  (source)