Ozone oxidizes bather waste; chlorine oxidizes bather waste.
When ozone breaks down (it has a short 15-30 minute half-life), it breaks down into hydroxyl radicals that are very short-lived powerful oxidizers; when chlorine breaks down by the UV in sunlight, it breaks down into hydroxyl radicals that are very short-lived powerful oxidizers. However, since chlorine is everywhere in the pool, these hydroxyl radicals are formed wherever there is sunlight in the pool. With the ozonator, only water passing through it gets exposed unless the ozone reaches the pool.
Ozone reacts with chlorine to form chloride and chlorate. One sees this very clearly in residential spas with ozonators, but in residential pools with ozonators most are so woefully undersized that one doesn't even notice the increased chlorine demand, but then what's the point of having such a weak ozonator anyway? If it's strong enough to show an increase in chlorine demand, again what's the point?
If one wants to use ozone in a residential pool, then one can have an ozone-only pool, BUT this requires a very powerful ozone unit (6% or higher ozone concentration in the injected air/ozone mixture) very fast turnovers (so large pumps and large diameter plumbing) and is not EPA-approved as a primary swimming pool disinfectant, mostly because it has not been shown to be safe (the EPA has tight rules for ozone in air so outgassing of ozone from an ozone-only pool is what they are worried about). Nevertheless, there are some very expensive high-end outdoor residential pools using ozone at 0.05 ppm in the bulk pool water, but you'd pretty much have to design your system to do this from the get-go.