The differences most noted would be the obvious 7” longer. Only an issue if weight and length are a concern. These rifles are heavy regardless. Also expect to loose 20-30 fps in velocity for every inch less barrel. Probably be around 200 fps less velocity with the 20” vs the 27”.
People prefer smaller calibers for several reasons. Ammo/component cost, recoil, don’t have the range needed to appreciate large magnums. 300 PRC can outshoot most peoples available ranges with ammo prices not crazy expensive like the the 300 Norma or 338 Lapua.
6mm is a different animal then others mentioned above. Many use this for competition. Makes a good flat shooting target round for paper and steel. Doesn’t have the knockdown power of the 300/338 if you want that. The 6mm are very accurate but can burn barrels faster than say a 6.5 cm
If you state what your intentions are (yes I’ve been following your other thread as well and it seems you don’t have a range in mind) we can help you more. If you plan to shoot steel to 1,000 yards then 6/6.5 gets my vote. Shooting steel to 1 mile then 300/338 get my vote. Beyond that other cartridges come into play. If you plan to hunt with this rifle then I favor the 300/338 for all ranges if your hunting something bigger than varmints like coyote or prairie dogs etc
As we mentioned in your other thread. If you buy the AXSR you can have every caliber you listed with the same gun.