Copper and aluminum are both effective materials for heat sink construction, but they have different requirements. If you want to know why, consider a great chef's kitchen.
Aluminum sure can move heat, if it has been done right. It very efficiently absorbs and transfers heat to it's environment and things that interact with it. This works great for bacon in the morning, and even for boiling water, but isn't so good for a large, thick filet mignon. That cold slab of beef sucks the heat right out of the aluminum, and there isn't any left to keep up the cooking. Many people who buy aluminum cookware have a lot of trouble doing steaks properly for this very reason. Aluminum has a low thermal capacity, and a very high thermal conductivity.
As such, aluminum just wicks heat away with little concern for anything else. It won't wick as much as copper, but it sure will move it quickly; Dumping it's capacity as soon as any heat leaves the sink, and quickly soaking up more.
Copper moves heat as well, even if it hasn't been done all that well. Copper very efficiently absorbs and transfers heat as does aluminum. It does it faster, as well. That said, copper has an incredibly high thermal capacity. That big fat steak just can't suck up all the heat that copper will hold on to, and this is where copper and aluminum differ in requirements. Copper won't readily dump all the heat energy it picks up, because it holds so much of it before it changes temperature to any great degree.
That leaves us with a problem. Copper needs help. Somehow, you have to remove all that heat from the copper, as it will just hold on to it otherwise. A copper heat sink can work much better than an aluminum one, but you have to either have loads of pipes and lots of fins and airflow, or you need peltier/water cooling with excellent transfer to help it out.
The thermal capacity of copper, when compared to an aluminum heat sink of the same design, completely removes the benefit of using copper in the first place without help. As a matter of fact, a poorly designed copper sink can be much worse than an aluminum model.
The best way to use the materials is being tried nowadays, and that is combining them. As with most good things, they work better together than apart.