I've shot several north american critters with a .300 Weatherby and a 30-06, so I believe I can offer an opinion.
It all depends on your bullet construction.
A lightly built bullet (such as a Nosler Ballistic Tip) will not penetrate as well when fired from the higher velocity gun at close range. I've personally experienced this with my .300 wby, where the bullets basically fragmented on impact, causing a large amount of superficial soft tissue damage but were somewhat lacking in penetration. The same bullet impacting at a lower velocity will be less likely to break apart and may actually penetrate better and be more effective.
However, you switch to a better constructed bullet, such as a Nosler Partition (A-Frame, Barnes, etc.), and everything changes. The heavily constructed bullet is designed to stay together and penetrate, even under the higher velocities. Using Nosler Partitions, I can honestly say that I do see a noticeable difference in the way game responds to a well-placed hit, vs my .30-06 at the same range. I believe that more energy gets transferred to the target. In my opinion, this is never a bad thing, provided you have adequate penetration.
So from my experience, yes velocity can make a difference in the impact on game, provided you're using a bullet that's up to the task. But at the end of the day, shot placement and penetration still reign supreme. The extra velocity of the .300 does not afford you any additional margin of error for shot placement. If you don't place your shot over the vital organs, and if the projectile doesn't actually penetrate to the vital organs, then it doesn't matter what gun you were using.