Looking at the numbers comparing .308 and .303Brit, (Military loads) you find the lighter .308 pill getting out of the gate a bit quicker, mainly due to developments in case design and propellants.. where the heavier .303 projectile carves a slightly higher curve, at about 1000yds their speeds have converged and the .303 actually has more retained energy.
Thats why the 7.62 NATO M118 SB and LR cartridges with 175 gr boat tail bullets are used for long range shooting by MG or Sniper.
The 150 gr .30-06 load had less effective range than the .303 in indirect MG fire so they developed the .30 M1 ball and matchgrade with bullets of 170+ gr.
They settled on the 150-152 gr M2 Ball for rifle use because the M1 Ball exceeded the safety limits of the rifle ranges in use in those days. That and recoil of the heavier bullet loads contributed to shooter fatigue.
At normal infantry ranges where the individual soldier is likely to score a hit on a mansize target the lighter and faster bullet will have the lowest maximum trajectory. The kill zone at a point blank sight setting is much greater if the velocity is even a few hundred FPS faster.
The difference can be made up for by better training and range estimation techniques.
For hunting the bullet choice, (weight, profile, etc) depends entirely on the game being hunted.