Both loads are adequate.
The CM is much more efficient. The case design is superior.
Short action is a bit easier to work the bolt (for young folks).
In rifles of identical weight, shooting bullets of identical weight, the 6.5 will have less recoil.
There is a plethora of 6.5 bullets available, for example, Hornady lists 22 6.5 bullets, while listing 14 for the .270.
The CM can double very well as a varmint rifle, while you can certainly use a .270, it isn't a good choice.
Factory offerings for the CM far outpace the ones for .270.The fact of the matter is this. The .270 is an excellent cartridge, hampered by bullet selection. The 6.5 CM is an excellent cartridge with a wide selection of excellent bullets.