This article aimed to compare various mechanical properties of epoxy/graphene and epoxy/halloysite nanocomposites. Graphene nanoplatelets (GnPs) and halloysite nanotubes (HNTs) were used as fillers at different concentrations. The studied fillers were dispersed in the epoxy resin matrices. Elastic-plastic mechanical behavior modulation was observed utilizing the fillers' nanoparticles and carboxyl-terminated butadiene-acrylonitrile copolymer rubber-modified epoxy resin. The hypothesis of the possible preceding inter-particle gliding of the individual GnPs in the complex resin nanocomposite matrix during mechanical testings was also confirmed. Increased ductility (elongation at break increased from 0.33 mm [neat matrix] to 0.46 mm [1 wt% GnPs] [39% increase]) and plasticity of the GnP nanocomposite samples were observed. In contrast, the decreasing mechanical stiffness as reflected in the decreased Young's modulus of elasticity (from 3.4 to 2.7 GPa [20% decrease]) was found for the epoxy/HNT nanocomposites. The obtained dynamic stiffness of the investigated nanocomposites confirmed the complexity of the mechanical response of the studied material systems as a combination of the ductile and brittle phenomena.