Chemical reasons for choosing Salt Water or Chlorinated pool?
A salt water pool still has hypochlorous acid $(\ce{HOCl})$ as the active component so essentially it CAN be the same thing as a chlorinated pool. For a chlorinated pool the $\ce{HOCl}$ is directly added, whereas for a salt water pool the salt water is electrolysed to create $\ce{Cl2}$ which then reacts with water to form $\ce{HOCl}$.

However, there is indeed often a difference and the explanation is partly chemistry and partly chemical engineering. In terms of chemistry  the overabundance of chloramines $(\ce{NH2Cl})$ in chlorinated pools over salt water pools makes the difference. When hypochlorous acid, the active component in both, comes into contact with ammonia (from sweat, saliva etc) you get the formation of these chloramines:
$$\ce{NH3 + HOCl → NH2Cl + H2O} $$
Chloramines are the major reason for irritated eyes and skin.

Now the difference between salt water and chlorinated pools has a chemical engineering reason. Salt water pools have a constant generation of chlorine as opposed to a 'batch' delivery in the case of chlorinated pools. Therefore the control over the concentration of hypochlorous acid in the water is better, which also allows better control over the concentration of chloramines.

So to sum up: salt water pools allow higher control over the concentration of chloramines and therefore might be better, however, if this control is not properly applied, it could also even be worse!