Or a carbon nanotube heatsink! Very good conductor. The problem with carbon nanotubes is it is a one dimensional conductor. All the chemical bonds are in one direction.
This depends upon how you arrange the carbon nanotubes (actually, carbon nanotubes are a bad choice for a computer heatsink as the thermal expansion coefficient is rather mismatched)
I was at a seminar a month or two back where they were discussing the possibilities of using a Tungsten core CVD diamond fiber in a silver matrix to give excellent thermal conductivity and the right coefficient of thermal expansion. Admittedly this was with a p.f. of 0.8 which is arguably infeasibly high, but it shows excellent potential.
 As an interested observer; really it was nothing to do with me and I probably shouldn't even have got invited, but hey ho.
On the original question, a somewhat helpful analogy is to consider yourself as a battery (hear me out on this) the air around you to be a perfect insulator, and when you touch the wooden block (a fairly good insulator) not much current (energy) flows through it because it has a high resistance. Conversly aluminium being a good conductor when you touch that a large current (more energy) flows through it
(Assume that you, the lump of wood and the lump of aluminium have a common ground).
Ask yourself would a battery shorted by wood drain faster than a battery shorted by metal? And then you have a pretty good analogy.
The energy (heat) is leaving you faster to go into the aluminium block, so it feels colder when you touch it. The key point is that you are at a higher potential energy level -be that a higher voltage in my battery analogy, or a higher temperature than the two blocks, meaning that energy flows from you into them. If the block were at body temperature rather than 20 C then neither would feel cold.
According to CES (material selection software) wood has a typical thermal conductivity of 0.31-0.38 W/m.K along the grain and 0.15-0.19 across grain. And Aluminium ranges between 76 -235 W/m.K (this is such a broad range as we haven't specified which alloy we are refering to)
For the other question, if the two materials are at a thermal equilibrium, then they have the same temperature by definition. If on the other hand you expose two materials with different specific heat capacities to the same amount of energy the yes, the temperatures will rise by different amounts. In the case of wood versus aluminium it will actually take more energy to increase the temperature of the wood by a given amount than it will the temperature of the aluminium.
But as you practical experiences tell you, it is the rate at which the energy is leaving your body that defines how "cold" a material is.