Cavity quantum electrodynamics: difference between Rabi oscillations and vacuum Rabi oscillations? I believe the title is pretty self-explanatory: I understand both roughly, but am unable to make a clear distinction. 
We use a circuit formulation for which an LC circuit, being a resonator, is the cavity, while the two-level systems (qubit) is a JJC circuit (LC circuit with a Josephson junction instead of the inductor).
The Hamiltonian for the qubit is:
\begin{equation}
\hat{H} = \hbar \omega \Big(\hat{n} + \frac{1}{2} \Big) = \hbar \omega \Big( \mathbb{I}-\frac{\hat{\sigma_z}}{2} \Big) \equiv \hat{H}_{I}.
\end{equation}
In the simplest case we consider only the qubit.
In the circuit formulation, if the qubit is driven with some frequency, one models it by substituting $V \mapsto V_0 + cos(\omega_{d} t)$ and (after some math) finds a new term in the Hamiltonian:
\begin{equation}
\hat{H}_{II} = K cos(\omega _d t) \hat{\sigma_x},
\end{equation}
where obviously $\hat{H}=\hat{H}_{I}+\hat{H}_{II}$ and $K$ is some constant. 
So if one represents the qubit on the Bloch sphere, the driven Hamiltonian is composed of two terms: the one that is  $\propto \hat{\sigma}_z$ is a "Larmor" precession about the $z$ axis (always present) while the one that is $\propto \hat{\sigma}_x$ is a precession about the $x$ axis, present only for driven qubits, and called Rabi oscillation; and the condition for which we pick a Bloch Sphere that is rotating in a way that cancels out the Rabi oscillations is called Rotating Wave Approximation (RWA).
(Is this correct up to this point ?)
Than, in the complete case where one considers the coupled atom-cavity, the full Hamiltonian is the Jaynes-Cummings Hamiltonian
\begin{equation}
\hat{H}_{JC} = \hbar \omega _c \hat{a}^\dagger \hat{a} + \frac{\hbar \omega_a }{2} \hat{\sigma}_z + \hbar g ( \hat{a} \hat{\sigma}^+ + \hat{a}^\dagger \hat{\sigma}^-).
\end{equation}
The three terms are related to the cavity, the atom and the coupling and $g$ is the coupling factor. The terms $a$ and $\sigma$ are creation/annihilation operators for the photon and raising/lowering operator for the electron in the atom: this is good, but I'm not able to work out the matrix form of $\hat{a} \hat{\sigma}^+ + \hat{a}^\dagger \hat{\sigma}^-$.
I get the identity matrix $\mathbb{I}$, while it looks like it should be $\hat{\sigma}_x$.
This would, in turn, give a case much similar to the previous one... only this time the oscillations would be called Vacuum Rabi Oscillations.
And at this point, I'm full of doubts. What's the physical difference? Why "vacuum" ? The qubit Rabi-oscillates without being actively driven: does this mean that coupling it to the cavity acts like some sort of passive driving, "vacuum" driving hence the name?
 A: The OP pointed out some similarities in the Hamiltonian of the two cases. There is, however, a clear physical distinction between the two cases.


*

*Vacuum Rabi oscillations are induced by strong coupling of the transition the the light field/cavity mode. They are thus a property of the Hamiltonian (see this question for details and related regimes), where the coupling is strong. What does "strong" mean? For details refer to the linked question, I would just like to give some physical intuition. When you have a photon in a cavity mode, it has different pathways to go. It can either excite the atom or go somewhere else, e.g. be lost to the environment. Strong coupling means that it is most likely to go excite the atom. We can see a signature of this if we consider the typical spontaneous emission scenario, where the atom is initially excited. At weak coupling (the Wigner-Weisskopf regime), the atom transfers its excitation to the cavity mode and the photon then decays elsewhere. Overall, this gives you an exponential decay. In the strong coupling regime, however, the photon is more likely to return to the atom. This gives oscillations even for single photons due to the strong coupling, called vacuum Rabi oscillations.

*Rabi oscillations are population oscillations induced by a strong driving field. You can have this in cavities and circuits, but also in free space. Rabi oscillations are not caused by a property of the JC-Hamiltonian parameters, but by what state you put in for the field. If we look the interaction part of the JC-Hamiltonian given by the OP $H_\textrm{int}=\hbar g \hat{a}\sigma^+ + h.c.$ this corresponds to the situation where $g\langle\hat{a}\rangle\sim1$ or bigger (relevant quantity is the Rabi frequency). We can see that you do not need a large $g$ for that, you can simply have many photons by making $\langle\hat{a}\rangle$ (or $\langle\hat{a}^\dagger\hat{a}\rangle$).


This distinction also shows why vacuum Rabi oscillations are so interesting. Since they can happen for single photons, many interesting quantum phenomena can be seen there. Rabi oscillations caused by a strong driving field on the other hand can often be treated semi-classically.
