Keldysh formalism and Kubo formula

Matsubara formalism works for statistical systems in equilibrium, and Kubo formula corresponds to the linear response approximation when the system is weakly perturbed by a time-dependent interaction. If you want to go beyond these two approximations (equilibrium or weakly perturbed systems), you need Keldysh method or some other methods of your preference, as e.g. to work directly with the density matrix and not Green's functions. For instance, if you want to treat far-from-equilibrium systems you need better things than Kubo or Matsubara.

Also, at the heart of the Green's function methods is the hypothesis of adiabatically switching on the interaction, using a theorem of Gell-Mann and Low [Bound states in quantum field theory. Physical Review, 84, 350–354, (1951).]. If you want to go beyond this approximation, you also need Keldysh method. Apparently, this is a problem which has only recently been seriously discussed, so I will not comment much about it.

Perhaps you mistakenly said Matsubara for the analytical continuation of the statistical equilibrium methods (i.e. the true Matsubara method) to time-dependent problems of statistical mechanics. Then the two methods are equivalent. It is just a matter of convention and preference which one you prefer to use (either Keldysh or analytic continuation of imaginary time). The reason is that in Matsubara method of equilibrium systems you suppose the time to be pure imaginary, so taking both real-time (i.e. evolution) and statistics means the time variable goes smoothly to the complex plane $t\rightarrow t+i/k_{B}T$ (sketchilly speaking, more details can be found in this question).

Also, time-dependent quantum field problems can be attacked using Keldysh methods (I mean, problem discussing only one particle, not statistical mechanics). But this is completely unrelated to Kubo or Matsubara approximations, so I guess your question was not about that possibility.

Finally, (and I'm less certain about that) I feel that quantum statistics can be discussed only in the Keldysh formalism. The reason is that the Green's function approach uses the classical statistics (I call classical both the fermionic and bosonic gases), i.e. the thermal equilibrium. This is the condition under which the replacement $e^{iHt}\rightarrow e^{H/k_{B}T}$ between quantum field and statistical fields works. Beyond this Boltzmann statistical weighting, I fear you need Keldysh or more powerful methods.