What exactly is the difference between a quasiparticle and composite particle in QFT/particle physics? I'm specifically thinking about quasiphotons as the superposition of photons and other interacting particles when light is traveling through a medium.  I've been told there's a fundamental difference between this and composite particles such as protons and neutrons.  But what is the difference exactly?  They're both the result of other particles/quantum fields interacting and wavefunctions interfering with each other, right?
Im an effort to figure this out, I've been looking up quasiparticles, trying to find an actual formal definition, but I just found a bunch of vague descriptions of emergent phenomena, mostly in classical physics, such as phonons and holes (i.e. electrons and holes in a semiconductor).  Those sound only vaguely related to the idea of a quasiphoton as the superposition of other particles (or, technically, the superposition of their wavefunctions) in that they're both emergent phenomena.
So, is there even a formal definition of quasiparticle, at least on the context of QFT and/or particle physics?  Or is it just some vague term used to describe anything that's sort of like a particle, but not actually one?
But in that case, what's the formal definition of "particle" in QM, QFT, and/or particle physics?  Is it that particles are Fourier transforms of wavefunctions (i.e. wave-packets), while quasiparticles are only superpositions of wavefunctions?  That doesn't seem quite right though, because I see no reason a superposition of wavefunctions can't also be the Fourier transform of another wavefunction. If anything, if I understand correctly, in QM the whole point of the FT is to convert between the position and momentum spaces anyway, and I see no reason we couldn't do that with a quasiphoton.
By the way, I've already seen this question: Is differentiating particle and quasiparticle meaningless?
but the single answer seems to constantly go back and forth between "yes they're the same" and "no they're actually not", without coming to a definitive answer or making it clear why there isn't one (other than saying they're the same mathematically, but have different physical properties, which makes no sense, as the entire point of the math is to model the physical properties)
I also looked at this one: Is a quasiparticle just an eigenstate of the Hamiltonian?
but the single answer uses a lot of notation I'm unfamiliar with so I don't understand most of it (I've mostly self-studied QM on and off for the past few years and I'm only just now finally taking an actual class that will help me understand more of the math, but it's only just started).  I'm hoping for a more qualitative/conceptual explanation, but that still makes clear the specific differences.
Alternatively, if that's not reasonably possible, could you stick to math that someone with an engineering degree but who hasn't formally studied a lot of advanced physics would be familiar with? Specifically, I've taken all required undergrad math courses up through diff eq, advanced linear algebra, and statistics, as well as the standard 3 semesters of university physics, as well as electrical engineering courses on electromagnetics and introductory semiconductor physics.  What QM I know, aside from the basics covered in physics 3 and the semiconductors course, is primarily from watching lots YouTube videos from reputable physics education channels, which got me familiar with the high-level concepts but left out most of the mathematical details, as well as posting hundreds of questions about QM on Quora over the past 3-4 years, asking specific questions to a friend with a master's degree in physics, and doing about half of Brilliant.org's "Quantum Objects" course.
 A: 
What exactly is the difference between a quasiparticle and composite particle in QFT/particle physics?

The term "quasiparticle" is often used in condensed matter physics to describe particle-like excitations of a many-body system. For example, an electron is a particle, but when it traverses solid matter it interacts with other electrons and atoms and can develop a lossy "self energy." It interacts, but still retains some of its particle nature, so we call it a "quasiparticle."
Sometimes the term "quasiparticle" is used generically to describe any excitations that can be written in a quadradic Hamiltonian like:
$$
\hat H \sim \sum_{\vec p}\epsilon(\vec p)c_{\vec p}^\dagger c_{\vec p}\;.
$$

If you want to hear from an authority, there is no greater authority than AGD:

"In other words, it is assumed that any energy level can be obtained as the sum of energies of a certain number of "quasi-particles" or "elementary excitations," with momentum $\vec p$ and energy $\epsilon(\vec p)$ moving in the volume occupied by the system.


A composite particle is a particle that is a bound state of other elementary particles. An elementary particle has no internal structure, but a composite particle does have internal structure and can be "split." For example an atomic nucleus can be thought of as a bound state of quarks (where the quarks are elementary).
A: The difference can be explained without using formal complications, Fourier transforms, etc.
First, a composite particle is a particle made up of more than one quark. They are something real, something tangible, "touchable". For example, a proton.
However, a quasi particle is not tangible, IT IS NOT REAL. That is the fundamental difference, one is real, something concrete that you can touch, the other is not. Quasiparticles are mathematical models to simplify the calculation in certain problems. Quasiparticles aren't physical real particles.
Some simple examples:

*

*The "quasi particle of the electron". When you want to analytically study the behavior of an electron in a lattice, something very useful is to assume that the electron has a mass different from its real mass. This simplifies the force interaction of the electron with the ions of the lattice.

*"Hole of the electron". When studying electrical conduction, the "hole" is not a real thing. It is not a particle that you can catch, it is precisely the absence of an electron. But studying the hole AS IF it were a particle makes the calculation much easier.

*Phonons. To study the movement of thermal energy in a crystal, you can study the vibration of each atom, one by one. This is too much work. You can "group" many atoms that vibrate in phase and call that group a "phonon". This makes the calculation much easier. In this video they put visual examples that will help you: https://www.youtube.com/watch?v=_axrpVnGHpk
Regards :)
