Why is the power supply terminal is not connected to ground in this bridge rectifier?
Both circuits output rectified unsmoothed DC, and if you plot the voltage across R1 and R2 you will find that both waveforms are absolutely identical, looking like this:  

  

However you are not plotting the voltage across R1 and R2. You are plotting the positive output of the rectifier against ground. In the first case ground is tied to the incoming AC, causing the weird output you see. In fact, grounding the other side of the AC supply would result in the exact same effect:  

  

If you are ever building a real world power supply, ground the negative DC output of the diode bridge (like on the right of your schematic), as grounding the input of the bridge is not only useless but will short out a diode if you also ground the negative output of the bridge.