Please excuse me, for any inability in my way to frame the question. I myself had a hard time making myself understand what was the doubt I really had :P. Also, I'm studying heat transfer at Undergraduate Level.
I first really need to make sure I understand the concept of thermal conductivity correctly or not. This is what I understand about thermal conductivity:
Thermal conductivity as defined in many books is a measure of a material's ability to conduct heat. A material with a high thermal conductivity will propagate heat faster within it. It tells about the speed at which thermal energy travels in a medium. So, suppose I have two identical rods- A and B made up of different materials such that thermal conductivity of A > thermal conductivity of B. Both the rods are initially at the same temperature and are brought in contact at the same time with identical heat source and heat sink. The heat transfer is one dimensional.
With the knowledge that thermal conductivity of A is greater than thermal conductivity B, I can only conclude that heat travels faster in A than in B, from one end to another. No conclusions can be made about the heat taken from the source by rod A and rod B (like I cannot tell by just a knowledge of thermal conductivity that which one of rod A and rod B will take more energy from source in any given time). Earlier I used to think that the one with a higher thermal conductivity will take more energy from the source in any given time.
Are there any other corollaries that I can make, if I know a material has a higher thermal conductivity? Corollary like if it conducts heat well, it will store less (I don't think so but..).
This is an excerpt from the book I'm referring to on the topic thermal diffusivity:
The sentence in the green - isn't that what thermal conductivity tells, about how well thermal energy propagates or diffuses into the medium?
The sentence in blue - just can't make sense.
Edit: Even though my question mentions 'rods', a more general explanation for difference between k and $\alpha$ will be appreciated, which is applicable to solids, liquids and gasses. As pointed out in one of the answers, solids have high thermal conductivities and high thermal diffusivities implying a solid conducts heat well and stores less. But this is not the case always a material can conduct less energy and store less also. Consider water and air for example, water has a higher $k$ than air, which means it conducts well but it does not store less (contrary to what solids do, they conduct well and store less), because water has a higher specific heat than air.