When is quasiparticle same as elementary excitation, and when is not? Can anyone shed light on the comparison between these two concepts?
 A: Dictionary for this answer: Excitation = particle; collective excitation = quasi-particle. 
Short answer:
Elementary particles are never quasi-particles, by definition of elementary. This does not mean that what now it is thought as an elementary particle could be a quasi-particle of entities to be discovered. 
Mathematical answer: 
Elementary particles are those that correspond to irreducible representations of the Poincare group.
Physical answer: 
Quasi-particles require the existence of an external medium or fields, whereas elementary particles do not. For example, phonons require a solid or a fluid to exist (they are collective modes of the atomic lattice vibration), likewise pions require a quark-antiquark sea. These are not fundamental particles, in the sense that they need the existence of other particles. A closed notion is that of composite particle, for example a molecule is made of atoms which, in turn, are made of a nucleus plus electrons. The difference between quasi and composite particles lies in the fact that quasi-particle are though as collective excitations of many particles (usually of the order on the Avogadro number $\sim 10^{23}$, but there may be far fewer, but not tens), while composite particles are more like building blocks where each constituent may be an elementary particle—such as an electron— or another composite particle—such as an atom— (a molecule is usually made of a few or tens of atoms, an atom usually contains from a few to tens of electrons plus a nucleus). Nevertheless, the difference between both concepts is not sharp; for example, pions are somehow made of quarks and antiquarks, they actually are also collective modes (waves) on the quark-antiquark sea, being quasi-Goldstone bosons of the approximate chiral symmetry.  
The difference between elementary and composite particles is tied to the human knowledge at the time. At a certain point, it was though that nuclei were elementary, after that people realized that there were in fact more fundamental constituents (protons and neutrons), and later on quarks and gluons were discovered. 
A: In the context of condensed matter physics:
To make it short, and with the caveat that it is not a universally accepted definition, an elementary excitation may be called a quasiparticle if it is fermionic (e.g. dressed electron), and collective excitation if it is bosonic in nature (e.g. phonon, magnon).
But there is not clear cut and absolute divide between such terms, and you will not get into trouble for using them in a lax manner.
