Copper coatings with refined grains and smooth surface morphology were electrodeposited from electrolytes comprising a novel accelerator, the disodium salt of 4,4-dithiobenzene disulfonic acid (DBDA), with sodium chloride and polyethylene glycol (PEG) as inhibitors and 2-aminobenzothiazole (ABT) as a leveler. It was found that the morphology of the coatings strongly depends on the presence and type of additives. In the presence of only DBDA and NaCl, large crystallites are formed, whereas the addition of PEG and ABT significantly decreases their size, and the most fine-grained, smooth, and defect-free copper coating is formed with the combined use of all additives. To establish the correlation between the observed morphology and the kinetics of the additive-assisted copper electrocrystallization in the proposed electrolytes, the nucleation mechanism and its parameters were determined by transient electrochemical characterization. The extended nucleation model, which takes into account not only the copper deposition but also the electric double-layer charging and hydrogen reduction process, was used to establish the electrocrystallization kinetics in the presence of the additives. The results of such kinetic analyses can help to explain the morphological effect. By using the chronopotentiometry method, it was found that the addition of the disodium salt of 4,4-dithiobenzene disulfonic acid with chloride ions gives a catalytic effect, while the sequential introduction of polyethylene glycol and 2-aminobenzothiazole provides an increasing inhibitory effect. Voltamperometry and chronoamperometry tests showed that the process is controlled by the diffusion of ions to the growing three-dimensional cluster of a new phase. The introduction of additives into the electrolyte slows down the copper electroplating process at comparatively negative potentials and increases the probability of transition from instantaneous to progressive nucleation. Moreover, the rate of the process and the density of nucleation active sites increase (up to 2–3 times) with the addition of DBDA but decrease significantly (up to 5–8 times) in the presence of PEG and ABT, which additionally confirms their catalytic and inhibitory effects, respectively, and explains the significant smoothing effect on the morphology of the Cu-coatings.