UV sterilizes are generally used for the purpose of killing bacteria and viruses. It does this by a photochemical reaction which causes parts of the DNA molecule to dimerise (forming a sort of "plastic") this stops it from reproducing.
Ozone breaks carbon - carbon double bonds. These occur in many nasty organics you don't want floating around your tank, and can sometimes make them easier to skim out.
In air, UV light with a wavelength of less than 240nm causes photodissociation of oxygen:
O2 ---> O + O
These O atoms combine with oxygen molecules to form ozone:
O + O2 + M ---> O3 + M
M is an inert substance (N2 gas) which absorbs the extra energy created by the reaction, and prevents the ozone from spontaneous decomposition back to oxygen.
I guess this would apply to dissolved oxygen within water as well. But there may well be other factors which inhibit it, so it's hard to say really.
Stability issues is not necesarily true. Ozone is stable, it's just that it is a powerful oxidiser, which means it reacts quickly. This is what the carbon is for, it provides a surface for the ozone to react with, to produce CO2 and O2.
Layton