Disclaimer: I am a developer of the Fleur code. I hope that I don't put too much bias into this answer. At least I try to not do that...
When you want to solve the Kohn-Sham equations you have the problem that the potential has singularities at the atomic nuclei. There are different ways to deal with this. On the one hand you can make the pseudopotential approximation to get rid of the singularities and make descriptions of the Kohn-Sham states with simple basis sets possible. On the other hand you can make use of a sophisticated basis set to get a representation on the basis of a potential including the singularities.
Codes like VASP or Quantum Espresso use the pseudopotential approach in combination with plane waves and projector augmented waves. The advantage of this approach is its simplicity in terms of parameters the user has to control and also in terms of development effort (with plane waves many expressions in the code are not too complex). The disadvantage is that the usage of pseudopotentials is an approximation, though a good one. Nevertheless it is difficult for the user to judge whether a result is affected or distorted by this approximation.
The all-electron full-potential linearized augmented-plane-wave method (FLAPW) on the other hand uses a very sophisticated basis set. It is used (with different flavors) in codes like Wien2k, Fleur, Exciting, Elk, HiLAPW, and a few more. Like plane waves the LAPW basis is systematically convergable. With this basis and the all-electron description you have a tool to obtain the DFT answer to a given problem and with a given XC functional in a controlled way with very high precision. You can produce reference results against which you can benchmark the precision of other approaches. The downside is that the LAPW basis is controlled by more than a single parameter like what you have in plane-wave basis sets. The complexity of the basis is a high entry barrier for new users of such codes and is also connected to more effort that has to be put into calculations and the development of the code.
The use cases of FLAPW codes (and other DFT codes) of course depend on their individual feature set (which is probably smaller if you have to put more effort into the development of new features). In general the strengths are wherever the results depend on details of the description, for example if tiny energy differences are involved. The all-electron description also allows for a rather direct evaluation of quantities that depend on the core electrons.
Your observation that FLAPW codes are used less frequently I can only answer with my subjective observation for the Fleur code. Within the last years we worked a lot on the user friendliness of the code, the user documentation, and the stability. This is an ongoing process but we already see rising interest in the code. It may very well be that the popularity of the different codes also depends a lot on how much effort the development teams can put into such aspects. I have the impression that since a few years the awareness of the need for such development efforts grows. Not only for the Fleur code but for all DFT codes and scientific software in general. I think DFT codes with very large user communities started working on such things earlier.
