We present a comparative study of the strain relaxation of GaN/AlN short-period superlattices (SLs) grown on two different III-nitride substrates introducing different amounts of compensating strain into the films. We grow by plasma-assisted molecular beam epitaxy (0001)-oriented SLs on a GaN buffer deposited on GaN(thick)-on-sapphire template and on AlN(thin)-on-sapphire template. The ex-situ analysis of strain, crack formation, dislocation density, and microstructure of the SL layers has established that the mechanism of strain relaxation in these structures depends on the residual strain in substrate and is determined mainly by the lattice mismatch between layers. For growth on the AlN film, the compensating strain introduced by this film on the layer prevented cracking; however, the densities of surface pits and dislocations were increased as compared with growth on the GaN template. Three-dimensional growth of the GaN cap layer in samples with pseudomorphly grown SLs on the AlN template is observed. At the same time, two-dimensional step-flow growth of the cap layer was observed for structures with non-pseudomorphly grown SLs on the GaN template with a significant density of large cracks appearing on the surface. The growth mode of the GaN cap layer is predefined by relaxation degree of top SL layers.