They work differently, Paolo.
The Balance slider is effectively two synchronised mono faders: one determines only how much of the left input channel passes through to the left side of the stereo output pair, and the other determines only how much of the right input channel passes through to the right side output.
The Power Pan control is an adaptation of the analogue mixing console type of stereo pan controls, as still modelled in ProTools and a few other DAWs. In this case left and right input channels each have an independent mono-to-stereo "pan pot" (panorama potentiometer) which shares its single-channel input audio in any user-set proportion between the left and right stereo output pair.
One important difference between these two types of panning is that when the Balance slider is set hard left, nothing from the right input channel passes through to either side of the stereo output pair; it is effectively muted. In contrast, when Power Pan is set hard left, both left and right input channels are mixed into the left side of the stereo output pair and nothing passes through to the right side.
I don't want to get into detailed comparisons of using these different types of pan control, but it is worth keeping in mind that when you have 2 input channels that, as a pair, contain recorded "dry" stereo information, such as in the VI Appassionata strings library, using Balance to place that sound somewhere in the stereo sound stage may not turn out to be quite as satisfying as when using Power Pan.