At inception the Pz III and IV were designed to be two tanks with different roles complementing each other: the PzIII was to be the anti-tank one with the high-velocity gun (the 3.7cm KwK!) and the Pz IV the support tank (7.5cm KwK short), to provide covering fire against anti-tank guns, machine guns, etc. Later on they evolved differently as the Pz IV was seen as having better adaptability but that's another story.
True, but it's strange that both tanks are around the same weight. The Panzer III is noticeably smaller and is armed with a smaller gun. Is it because of more armor?
Look at Panzerkampfwagen III and Panzerkampfwagen IV Yes, there were a few differences in armour schemes in the early models that would justify the weight difference for similar dimensions.