The results of systematic experimental studies of quantum oscillations of resistivity (ρxx), the Hall coefficient (RH), and capacitance (C) in GaAs and InxGa1-xAs quantum wells (QWs) with a simple electron spectrum and HgTe QWs with a complicated nonparabolic spectrum and strong spin-orbit interaction are reported. It is shown that for all the structures under study the oscillations of ρxx in a magnetic field (B) are completely determined by the oscillations of the density of states, the oscillation phase of ρxx(B) over the entire the filling factor (N) range is close to zero, which corresponds to the theory, while the oscillation phase of RH(B) is close to the theoretical value of π only for large N values and increases as N decreases. It is experimentally justified that such a behavior is not associated with approaching the quantum Hall effect regime. A striking result on the ratio of the amplitudes of the resistance and Hall coefficient oscillations is obtained. In GaAs QWs with a simple spectrum characterized by negligibly small Zeeman and spin-orbit splitting, the ratio of amplitudes is close to that predicted theoretically. In HgTe QWs, this ratio is very different and behaves differently in QWs with normal and inverted electron spectra. In HgTe QWs with a normal spectrum, it tends to a theoretical value with an increase in N, while for HgTe QWs with an inverted spectrum, it differs significantly from the theoretical one for all available N. It is assumed that such a difference in the ratio of amplitudes in GaAs and HgTe QWs is due not to the peculiarities of the energy spectrum of HgTe, but to the peculiarities of electron scattering due to spin-orbit interaction with the potential of the scatterers. This assumption is justified by analysis of experimental results obtained for a heterostructure with an In0.2Ga0.8As QW, which spectrum is very close to the GaAs QW spectrum, but characterized by a much stronger spin-orbit splitting value. It has been found that the positions of the resistance and capacitance oscillations, the difference between the phases of the resistance and Hall coefficient oscillations, and its N dependence are close to those observed in GaAs QWs. At the same time the ratio of the amplitude of the resistance oscillations to the Hall coefficient oscillations and its N dependence differ very strongly and they are close to that observed in HgTe quantum wells.