We investigate localization and band structure for quasistatic elastic waves in two-dimensional viscoelastic composites consisting of circular inclusions within a host matrix. Using the analytic continuation method of homogenization, we demonstrate a transition in the spectral properties of a key self-adjoint operator G that mirrors Anderson localization behavior. As inclusion size increases in disordered arrangements, the eigenvalue spacing distribution transitions from Wigner-Dyson to Poisson statistics, accompanied by increased eigenstate localization. In periodic arrangements, we observe clear band structure through sharp resonances in the spectral measure and effective complex shear modulus with mobility edges separating extended and localized eigenstates. Moreover, the phase of the complex shear modulus undergoes sharp switches from near 0 to 180 degrees at resonant frequencies, signaling transitions from elastic energy-storing response to a regime characterized by a negative effective shear modulus. These features disappear in disordered systems, where resonances in the spectral measure broaden, eigenstates and physical fields are typically more localized and associated mobility edges are less pronounced. Our results reveal how composite microstructure governs wave transport properties through an interplay between geometric order, inclusion size, and material impedance contrast, providing insights for designing materials with tailored elastic wave propagation characteristics.