Accessible system where specific heat and thermal conductivity vary in different ways? I'm teaching undergrad thermo this semester and to my surprise several students are having trouble conceptualizing heat capacity and thermal conductivity as different properties; they can apply them in equations just fine, but they are baffled as to why they are treated as distinct properties.
I'm going to try a different verbal track today, but I would love to be able to give them an example system where both quantities can be calculated from a microscopic model and they show different functional dependence.
Alas, this is not a question that I have asked myself and I don't have an answer.
For the ideal gas I keep finding expressions like 
$$ \kappa = \left( \frac{n\bar{v} \lambda}{s N_A} \right) c_v\,,$$
but I haven't followed the derivation closely enough to know if it continues to hold once rotational and vibrational modes are excited.
In my ideal world the system would admit a description that 3rd year undergrads could follow in detail.

I don't find an answer to this question in Are specific heat and thermal conductivity related?.
 A: Description of the difference
I will come at this from my perspective where I can't possibly imagine how thermal conductivity and specific heat capacity are not distinct properties. Maybe it will help. 
Specific heat capacity is a measure of how much energy can be stored in a system. We know it is related to the degrees of freedom for the molecules and it is really a property of the molecules. 
Thermal conductivity is a measure of how effectively collisions transfer temperature (and viscosity is a measure of how effectively collisions transfer momentum, diffusion coefficient, mass). This makes thermal conductivity a property of the fluid (as opposed to a property of the molecules). 
Derivation of $\kappa$
I will outline the derivation of the expression you have above as it is given in the intro chapter of Introduction to Physical Gas Dynamics. The derivation is set up for any generic property (mass, momentum, energy, etc):
Consider the 1D variation of a mean molecular quantity $\tilde{a}(x)$ which varies only in the $x$ direction. When a molecule moves across the plane given by $x_0$, it carries with it the value from the location of it's last position/collision, $\tilde{a}(x_0 - \delta x)$ when moving from the left of the plane to the right, and a value of $\tilde{a}(x_0 + \delta x)$ when moving from the right to the left. The distance from the plane, $\delta x$ is very close to the mean free path $\lambda$ such that $\delta x = \alpha_\tilde{a} \lambda; \alpha_\tilde{a} \approx 1$. 
They further assume that to a first order approximation, the number of molecules crossing the plane per unit area per unit time from either side is proportional to $nC$ where $n$ is the molecular number density and $C$ is the average speed of random molecular motion, both evaluated at $x = x_0$. This means that the net amount of $\tilde{a}$ is:
$$ \Lambda_\tilde{a} = \eta nC\left[\tilde{a}(x_0 - \alpha_\tilde{a} \lambda) - \tilde{a}(x_0 + \alpha_\tilde{a} \lambda)\right]$$
where $\eta$ is a constant of proportionality. Then a Taylor series expansion where only first-order terms are retained is:
$$\Lambda_\tilde{a} = -2\eta \alpha_\tilde{a} n C \lambda \frac{d \tilde{a}}{d x}$$
Okay, with all that out of the way, they then develop expressions for the diffusion of momentum and mass by comparing the expressions to the classical expressions. For thermal conductivity, the quantity being diffused is energy (heat) and they consider the internal structure: $\tilde{a} = \tilde{e} = (\xi/2)kT = (\xi/2)m RT = m c_v T$. This gives:
$$ q = -\kappa \frac{d T}{d x} = -2\eta\alpha_T n C m c_v \frac{dT}{dx} $$
And equating the two coefficients:
$$ \kappa = \beta \rho C \lambda c_v $$
where $\beta$ is a new constant of proportionality. This is effectively what your expression is.
tl;dr
Heat capacity is a measure of how a molecule can store energy. Thermal conductivity is a measure of how efficiently energy is transferred during collisions. 
Heat capacity is a property of the molecule (it depends only on the molecular structure).
Thermal conductivity is a property of the fluid (it depends on how the molecules interact with one another during collisions).
The expression you have shows that $\kappa \propto c_v$ so they actually vary pretty much the same way. And it is based on the internal structure, so it holds as other modes get excited.
Simple analogy
And maybe a simple analogy would work:
You have a wallet that has some slots to hold money. I have one also. Maybe my wallet has more slots than yours so I can store more money. This is a specific financial capacity.
You know I am just rolling in money cause I chose engineering as a career and times might be a little tough for you since you picked physics. So every time we pass in the hallway, you ask to borrow "some money," whatever I can spare. 
When I hand you money, I can hand you a whole bunch of bills, or only a few bills. If I hand you a bunch, my wallet slots will empty pretty quickly. If I hand you a little, it will take longer. As you take my money in each passing, your wallet slots will fill up (and maybe become totally full, and then you won't ask for money anymore). This is financial conductivity. 
Clearly how efficiently I can hand you money depends on how much money I have and how many empty slots you have in your wallet. So this financial conductivity depends on the money itself and your ability to store what I give you. 
Now, if our wallets were molecules and the slots our energy modes (and so our heat capacity), then our money would be energy. Every time we cross paths, I hand you some energy and you start filling up your modes while I deplete mine. How rapidly you fill and I empty at each crossing is the thermal conductivity. 
Hopefully that story would make it clear that while they show similar behavior, it's two very different properties.
