Difference between heat capacity and entropy?

$$dQ = T \ dS \tag1$$ $$dQ = C \ dT \tag2$$

Interesting, right? In $(1)$, the whole $T$ multiplies the infinitesimal $\frac{\text{J}}{\text{K}}$. In $(2)$ it's the opposite: the whole $\frac{\text{J}}{\text{K}}$ multiplies the infinitesimal $T$.

But you hinted that you knew that yourself already. Let's cut to the chase: both are different beasts entirely, just like heat and torque are not related just because they carry the same unit (joules are newton-meters, right?).

However, if you still want a defining difference between them, other than "they're just different", I'd give you this:

Entropy by itself is not useful and cannot even be measured. What is useful are changes in entropy, or how it differs from one state to the other. In this sense, it's akin to internal energy and enthalpy, for which only relative values matter. Heat capacity, on the other hand, can have its absolute value determined experimentally, and it won't depend on a reference value like entropy does. Its absolute value is immediately useful, if you will. In this sense, it's akin to pressure and specific volume, for which absolute values make sense.

If you consider a constant volume transformation, the corresponding specific heat will be defined as:

$C_v(T) \equiv \left( \frac{\partial U}{\partial T}\right)_{N,V}$

Now, it is not forbidden to use Leibniz rule for the decomposition of partial derivatives and for instance:

$\left( \frac{\partial U}{\partial T}\right)_{N,V} = \left( \frac{\partial U}{\partial S}\right)_{N,V} \cdot \left( \frac{\partial S}{\partial T}\right)_{N,V} = T \left( \frac{\partial S}{\partial T}\right)_{N,V}$

Which means that $C_v(T) = T \left( \frac{\partial S}{\partial T}\right)_{N,V}$

Hence, $C_v$ and $S$ are definitely two different things. In particular, the specific heat contains some (partial) information about the entropy of the system (and its possible variation under some constraints) but not all of it.

Hence in term of heat exchanged, we know that:

$\delta Q = TdS$ upon expanding $dS$ as total differential, we thus have one possible reading of the heat exchanged (at fixed number of particles):

$\delta Q = T \left( \frac{\partial S}{\partial T}\right)_{N,V} dT + T \left( \frac{\partial S}{\partial V}\right)_{N,T} dV = C_v(T) dT + T \left( \frac{\partial S}{\partial V}\right)_{N,T} dV $

The second term $T \left( \frac{\partial S}{\partial V}\right)_{N,T}$ is what any specific heat function (regardless whether we look at constant volume or pressure) will always miss and is ultimately related to the thermal expansion properties of the material.

As a partial answer (a complete answer would require more time and more knowledge about classical thermodynamics on my part): As you thought, these are two entirely different objects.

The heat capacity is a material-dependent object that - as you say - measures the difference in temperature when energy is absorbed by the material. It's object and process independent: If you add this amount of heat, you'll get that increase in temperature. EDIT: By "process dependent", here, I mean that the processes can be either reversible or irreversible. The capacity DOES of course depend on overall constraints: Leaving the volume constant or the pressure gives you a different heat capacity.

The entropy change on the other hand is a material-independent quantity. The entropy change is proportional to the heat transfer in a reversible process (at constant temperature! Otherwise you have to account for the temperature difference). However, most processes are irreversible, so the quantity is process dependent.

If you look at the statistical interpretation, the difference becomes more clear: The entropy counts the number of specific states that the system can be found in, given the thermodynamic parameters known. Since temperature is something like the averaged kinetic energy, the heat capacity loosely counts the number of degrees of freedom in the atom (heat is randomized kinetic energy that needs to be transfered to the material: The more degrees of freedom, the less energy every degree needs to take on when energy transfered to the material, i.e.: more degrees of freedom, higher heat capacity).