In projectile physics, F=ma. Rearranging that, F/m=a. The force of the burning powder divided by the mass of the bullet equals the acceleration of the bullet. Let's assume force is constant (the two cartridges have the same measure of the same powder); a lighter and heavier bullet both are given the same amount of energy at ignition, the lighter bullet, having less mass, will however have greater acceleration.
Now we add in friction between barrel and bullet. The coefficient will also be roughly the same, resulting in the same force of friction on the two bullets; however, due to its lighter mass, the lighter bullet will be slowed more by the barrel than the heavier bullet. Thus, given the same impulse through the same barrel, the heavier bullet will have more muzzle energy than the lighter one when it exits. To compensate, a greater impulse must be given to the lighter round (using a bigger powder charge) to give it the same muzzle energy.
It's also a matter of space. You take two FMJ bullets of similar nose shape. They're made of the same material. Yet one is heavier. The only way it can be so is if the heavier bullet is larger, and when bound to a certain diameter and overall length, that extra mass is in the cartridge, leaving less space for powder and for ignition. To give the same peak pressure according to ideal gas laws (pressure increases as volume decreases at a constant temperature and amount of gas), which produces the same impulse, the powder charge must be further reduced for the larger bullet over the smaller one.
