I believe there are a couple key differences
One is in how the selection is made. Another is "Adjustment Layers." You can create a Hue/Saturation Adjusment layer, but not with Replace Color.
Those difference help you decide when to choose between the methods. The actual hue/saturation adjustment and result should be the same in either case.
Replace Color creates the selection by clicking colors with the eye dropper. You can add/ subtract from the selected area via the +/- eyedroppers.
The other method requires a selection tool or creating a mask/ selection via painting or other method. There are many ways to select/ mask w/ Alpha channels, including building the mask based on hue/saturation content.
Here’s an instance where Replace Color probably won’t work. Say you have a picture that has a lot of a given color throughout, a field of purple flowers for example. You want to change the hue of a group of flowers in the center, but no where else. When you use the eyedropper to select purples in the center, if those same hues exist elsewhere in the picture, they will be select too. Using the selction method on the center flowers would be a better option.
However, if you wanted to effect all the purples in the picture, then Replace color might be a fast option. Even in this case, though, I’d probably choose Hue/Saturation over Replace Color. Mainly out of habit, but also because I prefer to use Adjustment Layers whenever possible.
Marshall