I believe that turbo charging was originally developed by the aviation industry to compensate for the effects of thin air at high altitude. The effect was to enable airplane engines to produce more power at altitude, thereby enabling an aircraft to maintain a good performance envelope at higher altitudes. In those days, the reason turbos were preferable to superchargers was because, unlike superchargers, turbos didn't cause mechanical drag on the engine -- which effectively robbed power to make power.
With the development of newer materials, better machining techniques and new compressor designes, I don't think the mechanical drag problem is as pronounced today as it was 50 years ago. Bottom line is that either turbo or supercharged engine will produce more power than same engine in naturally aspirated form. So while a naturally aspirated engine at 225 hp may loose, let's say, 25 hp at a certain high altitude, a turbocharged or supercharged engine at 340 may loose the same 25 hp but still come in at 315.