The investigation and construction of complex hierarchical structures from advanced composite materials through various additive manufacturing processes is developing rapidly worldwide and plays a particularly crucial role in the manufacturing and industrial sector. One of the most popular 3D printing techniques is Fused Filament Fabrication (FFF), which utilizes thermoplastic filaments as printing material. The mediocre mechanical performance of thermoplastic polymers has resulted in the rapid research and progress of advanced nanocomposite materials with the aim of improving the mechanical properties of complex structures. The objective of the current study is firstly to design and manufacture FFF 3D-printed hierarchical honeycomb and spiderweb constructs, using two advanced composite materials, polylactic acid reinforced with nanodiamonds (PLA/uD) and acrylonitrile butadiene styrene with carbon fibers as reinforcement (ABS/CF). At the second stage, there was an examination of the mechanical performance of these involute structures under experimental and theoretical tests. In particular, the compression behavior of the fabricated hierarchical honeycombs and spiderwebs was assessed by experiments along with finite element analysis (FEA) simulations. According to the results, the hierarchical cellular structures exhibited enhanced mechanical properties in comparison with the spiderweb structures. Specifically, augmented stiffness and compressive strength were observed. In addition, the increase of hierarchy in the honeycomb structures resulted in an improvement in the mechanical behavior in contrast to the spiderweb constructs. Finally, the convergence between the experimental and theoretical results was quite good for both hierarchical structures and materials.