Where is the basic difference of statistical mechanics with many-body physics? What are the systems which cannot be studied in statistical mechanics but in many body theory? After all we know interacting systems can be dealt with in statistical mechanics for example in studying the models of real gases, Ising-like systems by mean field theory etc.
The uses of this two theories are completely different.
Statistical Mechanics is used to see how by modelling the behavior of microscopic constituents you can predict the macroscopic phenomenas that you observe.
On the other hand Many Body Theory uses first principle techniques to see what happens microscopically when you have large no of particles in your system. Explaining the macroscopic phenomena are the second step here.
Because of their different techniques the applicability of these two theories are also different. For instance if we have a system with 20 atoms each with 2 valence electrons then if you use statistical mechanics the statistical fluctuations(uncertainty) in your results will be comparable to the actual results. Here you have to use Many Body Theory. On the other hand when you have 10^23 molecules, solving many body equations will become unimaginably tough even with largest supercomputers. There people use statistical mechanics.
I don't have an "official" distinction, but one difference is that statistical mechanics is used for systems with a very large number of particles, and is only concerned with quantities which are averaged over the whole ensemble. Many-body theory deals with smaller systems, and attempts to treat all the particles in the system, within some approximation. Another is that many-body methods are typically used to apply to problems where the density of states is low, and they describe individual quantum states of the system.