Like the .30/06 the .303 provides good performance with heavy bullets. The velocity is high enough with a 180 gr bullet for a flat trajectory out to 300 yards, but is low enough that the bullets perform well on game without failure. For game bigger than moose, the key is to find a good bullet and good bullets are easier to find in the '06's .308" than with the .303's .311". I believe Woodleigh makes such a bullet.
Most folks won't hunt bison or big bears with a .303, so it is of little consequence I suppose, but in Africa the .303 has a reputation as a wounder where the '06 has a reputation as a killer. I suspect the reason is that the Africans like the 220 gr round nose .30/06 but use the .303 with surplus ammo and it is often easier to shoot more accurately with a high quality .30/06 sporter than it is a Lee Enfield.
Having used both, there are situations where I would choose the .30/06 over the .303. A 220 gr bullet in the .30/06 has the velocity and trajectory of the 180 gr bullet in the .303. When I want a flat shooting reach out there shot, the .30/06 with a Nosler 165 gr BT at 2800 pretty much out classes the .303. If all this sounds a little negative, let me say that when loaded with 180 gr bullets, a 10 shot Lee Enfield in typical central Canadian bush country epitomizes what a hunting rifle needs to be.