They overlap at least, and they may even be synonymous, depending on who you ask. Specialists in different fields often end up borrowing each other's ideas and methods, and the lines between them often become blurred or found to be nonexistent in the first place. Refs 1, 2, 3, and 4 are just a few of the many examples. And by the way, the author of ref 1 is a Physics SE user.
To explain why the meanings of "many body theory" and "QFT methods" are so fluid, I'll give a broader perspective that includes relativistic QFT.
The broad class of models that includes models called "many body theory" and models called "QFT" can be characterized like this:$^\dagger$ instead of assigning observables to individual particles, observables are assigned to regions of space (in the Schrödinger picture) or spacetime (in the Heisenberg picture). In such models, individual particles don't carry their own observables, but we can have observables that detect particles in a given region of space(time). This approach is important even if the total number of particles is conserved, because particles (like fermions or bosons) are often indistinguishable — which is really just a cryptic way of saying that the model's observables are tied to space(time) instead of to individual particles.
$^\dagger$ In topological QFT (TQFT), "spacetime" is a misnomer because such models don't have a metric field, and observables may not be assigned to local (contractible) regions. TQFT is still a rich subject because we can consider the model on a whole class of different spacetimes simultaneously. This is also an interesting thing to do in non-topological QFT, but I won't go into that here.
Here are a few comments to highlight the diversity within this broad class of models:
It includes both strictly nonrelativistic and strictly relativistic models, and also models that are neither strictly nonrelativistic nor strictly relativistic. One example of the "neither" type is a variant of quantum electrodynamics with strictly nonrelativistic electrons: the number of electrons is conserved, but the number of photons is not.
It includes models where the total number of particles is conserved, and models where it is not conserved. (More generally, within a single model, the total number of a given species of particles may be conserved for some species and not for others.) This is different than the distinction between relativistic and nonrelativistic models. The total number of particles is conserved in relativistic models that don't have any interactions at all, and it's not conserved in many nonrelativistic models. Even in nonrelativistic models where the number of "fundamental" particles is conserved, we still often have emergent quasiparticles (like spinons and holons) whose number may not be conserved.
It includes models that assign observables to regions of continuous space and also models that assign observables only to discrete lattice points. The latter is not limited to condensed matter, and it's not limited to numerical calculations. In fact, lattice QFT is currently the most broadly applicable method we have for constructing relativistic QFTs nonperturbatively. A strict continuum limit may not always exist (unless we forfeit the interactions that make the model interesting), but we can still take the lattice spacing to be much smaller than the Planck scale, and that's close-enough-to-continuous for all practical purposes.
Many of the models that were originally conceived for condensed matter turn out to define relativistic QFTs when the parameters are tuned to make the correlation length much larger than the lattice spacing, and this situation has enriched both subfields. We can use techniques originally developed for relativistic QFT to study critical points and phase transitions in condensed matter, and we can use techniques originally developed for condensed matter to help construct new examples of relativistic QFTs — or sometimes just new and enlightening ways of constructing familiar examples.
Whatever we call them, interesting models whose observables are tied to space(time) are so diverse that any attempt to neatly classify them is doomed. The more we learn about the subject, the more apparent this becomes. Words like "many body theory" and "QFT methods" are little more than vague clues about some of the kinds of things the author/speaker might be referring to. We just can't rely on words like that to convey anything precise, unless the author/speaker tells us exactly how we should interpret them in that particular article/lecture.
References:
X.G. Wen (2004), Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons, Oxford
Zinn-Justin (1996), Quantum Field Theory and Critical Phenomena, Oxford
Abrikosov, Gorkov, and Dzyaloshinski (1963), Methods of Quantum Field Theory in Statistical physics, Dover
McGreevy (2016), "Where do quantum field theories come from?" (https://mcgreevy.physics.ucsd.edu/s14/239a-lectures.pdf)
