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The difference between the two pairs of cartridges (220 vs 22-250 and 308 vs 30-06) are simple: case volume and working pressure.
The 22-250 has a case capacity of 45 grains of water, and a working maximum pressure of 65,000 psi.
The 220 Swift holds 46 grains of water and a maximum pressure of 54,000 CUP.
The comparison may seem skewed in pressure, but that's the difference between a psi measurement and a CUP measurement. In reality, both rounds are as alike as peas in a pod.
With maximum pressure loads, the 22-250 will boost a 50-gr bullet to about 3850 fps. With the exact same bullet (with the same BC), the 220 Swift will get 4000 fps. Believe me when I say that there isn't a spit of difference between 3850 and 4000 fps at the muzzle. More importantly, those numbers might be reversed with two different rifles, with the Swift being a titch slower.
The difference between the 308 and the 30-06 is more dramatic in physical measurements. The 308 holds 56.0 of water while the 30-06 holds 66.0. Ten grains makes a big difference - except for pressure. Due to the age of some rifles (a century), the 30-06 is now loaded to a maximum of 55,000 psi, while the newer 308 (and its stronger rifles) can be loaded to 60,000. At peak pressure, with the same bullets, the two are also identical in performance. The 308 will get a bit over 2800 fps with a 150-gr bullet, and the 30-06 will get a bit over 2900. No real difference.
The reason that many target shooters use the 308 instead of the 30-06 (at ranges of 600 meters and less) is not velocity, but recoil. The 308 gets its performance with as much as ten grains less powder, so it has significantly less recoil than the 30-06. Less recoil= less flinching=higher scores. It's that simple.
(At much longer ranges, the edge goes to bullets with extremely high BC, and smaller cases can't launch them fast enough to take advantage of the BC. So 1,000-yard shooters often use mega-magnum cartridges to launch very heavy, very long, very streamlined bullets - and have to suffer the recoil.)
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