Polaron energy spectrum of AlAs/GaAs/AlAs and PbS/PbTe/PbS quasi two-dimensional heterostructure as a function of the wave-vector is investigated based on the adiabatic approximation and the Ritz variational technique. The electron interaction with confined,interface and half-space phonons is taken into account. A comparative analysis of the polaron energy spectrum in heterostructures with a finite and infinite breakdown of energy bands at the interface is performed by means of the Li-Lou-Pines variational method. The polaron density of states dependence as a function of energies with the values less than phonon energies is studied. The contribution of each phonon branch to the density of states is determined. A comparative analysis is made for two AlAs/GaAs/AlAs and PbS/PbTe/PbS heterostructures.