The dynamics of Newtonian nonelastic and viscoelastic droplets that impact an immiscible liquid surface were experimentally studied using high speed visualization techniques. The impact mechanisms of nonelastic droplets differed from those of viscoelastic droplets. The azimuthal instability seen along the rim bordering the nonelastic droplets was not observed during the impacting of viscoelastic droplets. The azimuthal instability is attributed to the Saffman–Taylor instability because of the viscosity discontinuity across the interface, and to the Rayleigh–Taylor and Richtmyer–Meshkov instabilities because of the density difference at the interface. The effects of the physical parameters, in terms of the Weber number, We, on the growth of the azimuthal instability were studied. The analysis revealed that the growth of the azimuthal instability increased the inertial force of the droplet upon impact. Moreover, surface tension-driven instability, known as the Plateau–Rayleigh instability, was also observed from impact of the nonelastic droplet, which was distinct from the viscoelastic droplets. The stabilizing role of the elasticity in the droplet impact was investigated using the elastocapillary number, Ec. For nonelastic droplets, the elastocapillary number is negligible, hence any disturbance could grow further and destabilize the liquid. However, for viscoelastic liquids, the Ec is significant due to the presence of elasticity, which prevents the growth of any disturbances in the liquid.