Yep, it's the d-d energy splitting, which is different for the different metal ions.

In this compounds with larger crystal field splitting exhibit colors with higher energy and shorter wavelengths. This corresponds to the blue color observed in [Cu(H2O)6]2+

Visible absorption, which is related to the colour of compounds, can originate from a wide variety of mechanisms, but all of them have in common the excitation of an electron from one energy level to a second level of higher energy than the first. The absorption bands observed in the spectra are of different types: d-d transitions, Charge transfer bands (MLCT and LMCT), Internal ligand transitions. The position of the absorption bands of d-d transitions give information about the type and characteristics of the ligands and the E.O. of the metal. In the specific case of transition metal complexes, absorption in the visible is originated when an electron is excited between two energy levels that are both d orbitals of the metal ion (d-d transitions). The absorption spectrum presented by transition metal ion complexes depends on the energy of the d orbitals, their degeneracy and the number of electrons distributed in them; In turn, these facts are conditioned by the oxidation state of the metal ion, the number and type of ligands and the geometry of the complex. In the absorption spectrum of Ni(II) complexes, such as the [Ni(H2O)6]2+ complex, the energy of the excited terms 3T2g , 3T1g(F), and 3T1g(P) can be represented graphically as a function of (delta)o , taking as zero energy that of the fundamental term 3A2g (Tanabe and Sugano diagrams). Consequently, we have 3 absorption bands. For a Cu2+ ion and the specific case of the [Cu(H2O)6]2+ complex in the Tanabe and Sugano diagram, there is only one possible transition and one absorption band. This explains the different color of both complexes.

They have different numbers of d electrons.

Copper(II) is d9 and will have significant Jahn-Teller distortions. Nickel(II) is d8 and won't.

The amount of energy required to bump an electron from the low energy orbitals to the higher ones isn't just a matter of the separation between the two. It also depends on which orbital it's coming from, where it's going, and what other orbitals are occupied. That's why simple octahedral Ni(II) complexes generally have three peaks in the visible spectrum.