Weight: heavy barrel is heavier.
Accuracy: I can't see the difference.
With my light barrel (and a 2 MOA Aimpoint), I shot a qualification score high into the expert range. Didn't top out the qual, but did close enough that it raised eyebrows.
Everyone has opinions, but mine is that a heavy barrel just adds weight.
As far as being concerned about heating, I ran 390 rounds though mine as fast as I could squeeze the trigger...... did it get hot? Yup. Still shot.