This study experimentally investigates the effect of projectile nose shape on the low-velocity impact response of sandwich panels with 3D-printed lattice and corrugated cores. The face sheets were manufactured from glass-fiber-reinforced polymer composites, and the cores were fabricated from polylactic acid (PLA) using the fused deposition modeling (FDM) process. Drop-weight impact tests were conducted at different energy levels using blunt, hemispherical, and conical projectiles. Force–time and displacement–time histories were recorded, the absorbed energy was obtained from the corresponding force–displacement curves, and post-impact inspections were performed to identify damage mechanisms. The results show that projectile nose geometry significantly influences contact conditions, stress concentration, and failure modes: blunt projectiles generally produce higher peak forces, whereas conical projectiles promote localized penetration and more severe localized damage. Regarding core architecture, corrugated cores exhibited higher initial stiffness under a considerable portion of the tested conditions, while lattice cores provided superior specific energy absorption at certain impact energy levels. Overall, the findings indicate that the relative advantages of each core type depend on impact energy and nose geometry, emphasizing that core selection should be guided by the intended performance criteria (initial stiffness versus energy absorption) and loading conditions.