We report here the synthesis, crystal structure, characterization and anticancer activity of a copper(II)-hydrazone complex, [Cu(MeBHoVa)(H2O)2](NO3) (for short, CuHL), against human breast cancer cells on monolayer (2D) and spheroids/mammospheres (3D). The solid-state molecular structure of the complex has been determined by X-ray diffraction methods. The conformational space was searched and geometries were optimized both in the gas phase and including solvent effects by computational methods based on DFT. The compound has been characterized in the solid state and in solution by spectroscopic (FTIR, Raman, UV-vis) methods. The results were compared with those obtained for the hydrazone ligand and complemented with DFT calculations. Cell viability assays on MCF7 (IC50(CuHL) = 1.7 ± 0.1 μM, IC50(CDDP) = 42.0 ± 3.2 μM) and MDA-MB-231 (IC50(CuHL) = 1.6 ± 0.1 μM, IC50(CDDP) = 131.0 ± 18 μM) demonstrated that the complex displays higher antitumor activity than cisplatin (CDDP) on 2D and 3D human breast cancer cell models. Molecular docking and molecular dynamics simulations showed that CuHL could interacts with DNA, inducing a significant genotoxic effect on both breast cancer cells from 0.5 to 1 μM. On the other hand, CuHL increases the ROS production and induces cell programmed death on breast cancer cells at very low micromolar concentrations (0.5–1.0 μM). Moreover, the compound decreased the amount of breast CSCs on MCF7 and MDA-MB-231 cells reducing the percentage of CD44+/CD24−/low cells from 0.5 to 1.5 μM. In addition, CuHL overcame CDDP with an IC50 value 65-fold lower against breast multicellular spheroids ((IC50(CuHL) = 2.2 ± 0.3 μM, IC50(CDDP) = 125 ± 4.5 μM)). Finally, CuHL reduced mammosphere formation capacity, hence affecting the size and number of mammospheres and showing that the complex exhibits antitumor properties on monolayer (2D) and spheroids (3D) derived from human breast cancer cells.