Genuine Tube-like sound that makes it more than your average Tube Screamer!
 
More gain! That was the overall goal. If you're not playing through a real tube amp, cranked for overdrive, it's difficult to get genuine tube overdrive sound. 
The original Tube Screamer pedals used an OP-AMP system, a basic circuitry that became popular for its “perfect amp” attributes. It had no distortion so it had to be overdriven or distorted by re-shaping the signal until it distorts (diode-clipping).  Clipping it a little bit gives you Overdrive. Clipping it a lot gives you Distortion.