How does complex ion differ from ligand complex?
To quickly pre-answer the question at the end of the body: Yes, the oxygen atoms around the chromium(VI) cause d-orbital splitting. However, chromium(VI) is a $\mathrm{d^0}$ system and thus the colour is not caused by d-electron transitions. Rather, as I highlighted in my permanganate answer, the cause of the colour is a ligand to metal charge transfer transition. For details, see there.



The question in the title basically refers to the difference between a complex such as $\ce{[CoCl4]-}$ and an ion such as $\ce{SO4^2-}$ or $\ce{CrO4^2-}$. These can be considered two extremes of a transition much like the extremes unpolar covalent bond and ionic bond when used to describe bonding.

All coordination complexes are nothing more than dative bonds, which in turn are nothing more than a special subset of weak covalent bonds. From a qualitative bonding point of view, there is really no difference between either compounds: For both you get orbital mixing, a bonding and an antibonding set of orbitals and an overall stabilisation of the system. And if you totally ignore energy differences and the like, you will find that each of the three compounds written up there have more or less the same MO scheme.

The difference from a theoretic point of view is the bond strength and the energy difference between the central atom’s and the ligands’ orbitals. For a complex ion, the orbital energies of the unbonded atoms are very similar to each other resulting in a large energy difference upon forming a bond and therefore a large difference between bonding and antibonding orbitals. For typical ligand (or coordination) complexes, the ligands have much lower orbital energies than the central metal and thus the energy gained in the bonding process is comparatively small. On this scale between $\ce{[CoCl4]-}$ and $\ce{SO4^2-}$, chromate(VI) would be somewhere just to the sulphate side of the axis’ centre.

What does this difference mean from a practical point of view? Well, one typical reaction of coordination complexes is the ligand exchange reaction: Add a better ligand into the solution and more often than not it will exchange with a worse ligand. Sometimes this results in a different geometry. $\ce{[CoCl4]-}$ can be synthesised by preparing an aquaeous cobalt(III) solution which contains $\ce{[Co(H2O)6]^3+}$ ions and adding excess chloride ions — the complex will form spontaneously. The reverse reaction is possible by adding silver nitrate which will precipitate the chloride ions, reforming the hexaaqua complex. Ligand exchange is simple.

The sulphate ion, as the extreme opposite example, has such strong $\ce{S-O}$ bonds that they will not ‘simply dissociate by offering a better ligand.’ (In fact, I don’t know if a better ligand even exists.) It is impossible to displace the oxygens to attach other ligands — and most notably, there is no good oxophilic compound known that could simply remove (i.e. precipitate) those oxygens. Note, however, that sulphuryl chloride $\ce{SO2Cl2}$ will readily react with water to form sulphuric acid (diprotonated sulphate) and hydrogen chloride, so one type of ‘ligand exchange’ is still possible.

Chromate(VI), as written earlier, is closer to sulphate than to the cobaltate complex. It will — much like permanganate — not take part in displacement reactions, the metal–oxygen bonds are pretty stable. Again note that an analogous chloride complex exists with chlorochromate $\ce{CrClO3-}$.

Note that not always will the ligand exchange I highlighted above occur easily. Ammonia should form a more stable complex with iron(III) than water. However, when adding ammonia to an iron(III) solution, iron hydroxides will precipitate rather than an ammin complex being formed. So the practical point of view is a really rough approximation that only works sometimes.