According to the tug driver, during pushback of the airplane from the gate, he turned the airplane early in order avoid an airplane at an adjacent gate. He further stated that he did not straighten the tug in time and impacted the airplane which resulted in substantial damage to airplane's nose landing gear support beam and a puncture in the pressure vessel. According to the pilot of the flight, during pushback the airplane overshot the centerline and was being corrected back to the line when he observed the tug continue aft and then heard a metallic sound. The pilot further stated that the pushback did not appear to be too fast and there appeared to be only minor directional control problems with the tug until the impact.