Your question is rather broad and the partial answers for the different aspects are quite complex. Maybe I can clarify some of them.
In the DFT developer community one typically distinguishes between accuracy and precision. Accuracy is understood as a measure for the agreement between simulation results and experiments. This is in the end limited by the XC functional. Precision covers for a given XC functional the agreement between the simulation results with a certain code and the theoretically possible correct result. I assume that in your question you actually want to know something about precision.
Here, FLAPW methods offer the chance to come very near to the theoretical optimum, because beyond the XC functional all other approximations in such methods can be controlled and reduced to have negligible precision effects. Pseudopotentials (and also basis sets that are not systematically convergable) are approximations that cannot be controlled in such a way. PW basis sets are systematically and very comfortably convergable. You actually only have a single convergence parameter there: An energy cutoff controlling the basis set size. For equation of states paremeters, we are on the way of publishing a reference dataset for the measurement of the achievable precision with specific DFT implementations together with results for different DFT codes / pseudopotentials. See https://arxiv.org/abs/2305.17274 for details. This can be seen as a successor to the "Delta project". In the supplementary part of that work you can also read for what kind of systems challenges for pseudopotentials are found.
For band structure calculations you should be aware that DFT codes are typically constructed to calculate the charge density. This only relies on the occupied bands. In FLAPW codes you therefore have "energy parameters" around which a linearized description of the Kohn-Sham states takes place. These are chosen to get a good description of the occupied states and typically also provide a very nice description a few eV above the Fermi energy. If you want to also consider higher unoccupied states there is the option to extend the LAPW basis with local orbitals, i.e., additional basis functions that can be tailored to represent such states. In the construction of pseudopotentials one also plugs in the requirement that certain states (those required for the occupied states and a few more) are nicely represented. I don't know if people have mechanisms of extending the precision of pseudopotentials for Kohn-Sham states far above the Fermi energy. I am also not aware of studies demonstrating where the limits are.
The charge densities from different methods differ. Especially the charge densities from pseudopotential codes represent artificial systems covering the valence electron states only. To some extend this may be overcome by using the PAW approach, but even this is typically used together with a frozen core electron approximation, so you will see differences from all-electron approaches when you take a close look. But, of course, you already see small differences within a class of methods, e.g., between FLAPW calculations from different codes and with different parametrizations or differences between FLAPW codes and other all-electron approaches. You will also see differences between different pseudopotential codes.
With respect to performance both, pseudopotential PW codes and also FLAPW codes, are limited by the diagonalization of the eigenvalue problem. This scales cubically with the system size. However, PW codes typically have a larger basis set and thus also larger matrices. For the representation of the valence Kohn-Sham states they only need a small fraction of the eigenfunctions of the matrices and for this often iterative diagonalization algorithms are employed. These are efficient when only few eigenvalues / eigenfunctions are needed as they allow to skip an explicit construction of the whole Hamiltonian matrix in terms of the basis functions. FLAPW codes on the other hand have smaller matrices because the basis sets are also smaller. Here the diagonalization is typically done by direct eigensolvers, implying the need to construct the matrices explicitly. Another aspect is that PW codes employing norm-conserving pseudopotentials only have to deal with simple eigenvalue problems, while FLAPW codes come with generalized eigenvalue problems. Finally, in FLAPW codes there are some other computational kernels that scale cubically with the system size. Overall, I assume that PW codes are typically a little bit faster, but that depends on the unit cell and type of calculation. If you have many valence electrons per volume, the advantage of the iterative diagonalization schemes becomes smaller.
With respect to calculation types, some calculations are possible in certain codes while they are not possible in other codes. For example, as far as I know, when you perform a calculation considering spin-orbit coupling (SOC), with QE you always perform a calculation for systems with noncollinear magnetism. This is computationally expensive. In FLEUR you can also calculate SOC for nonmagnetic systems (e.g. for the Rashba effect) or systems with collinear magnetism (e.g. for calculating the magnetocrystalline anisotropy). Another difference shows up when you perform calculations on thin film systems. In PW codes this can only be done in terms of periodic slab calculations, i.e., you introduce a large vacuum region in the unit cell. This increases the number of needed basis functions. In FLEUR one has the possibility to set up a 2D geometry which does not come with increased basis set sizes.
The difference in calculation modes, however, is only partially related to the comparison between PP-PW codes and FLAPW codes. Even within a certain approach different codes may offer different options on what type of calculations can be performed. To my knowledge, for example no other FLAPW code at the moment offers the 2D geometry that FLEUR offers for thin film systems. Also for different FLAPW codes the performance characteristics may be slightly different. You can think of FLAPW codes as codes employing some basis set from a family of basis sets. In Wien2k typically an APW+lo basis set is used. In FLEUR most calculations are performed with a conventional LAPW basis set.
As final points let me mention that independent of possible uses of these codes on laptops or workstations, both, QE and FLEUR, are part of the MaX collaboration. All codes in this collaboration have been refactored and tuned for extreme parallelization scaling on large supercomputers. When it comes to very large unit cells that require a large-scale parallelization of the DFT calculation, such codes may offer advantages in comparison to other codes of the same approach that might not have been developed for such use cases. Also, getting near to the achievable performance of such codes requires some experience from the user: The codes have to be compiled in a smart way and the employed parallelization scheme also has to be tailored for the specific calculation.