I do not own a 6,5 CM, but I am using 6,5×55 which is is basically the same thing. Two decades ago I was using 30-06 and 308 win calibers and then went for 6,5×55 as I was looking for something with a little bit less recoil due to health issues but with still enough punch to be able to hunt wild boar and red deer. For years I was pleased by the 6,5 mm, its accuracy - which is by the way better than 308 win at least in my experience, and also its somewhat lower recoil while still giving enough punch for me to dispatch even large game cleanly. It's excellent accuracy combined with a pleasant recoil makes it an extremely efficient cartridge for hunting light to medium game.
But comparing 308 and 6,5 mm side by side, I must make a few remarks.
First, barrel length. While the 6,5 mm is decent from a 20 inch barrel you still just can't shake off the feeling that you are walking right on the edge with regards to how much punch the cartridge packs, especially when hunting large game. 6,5 mm really likes somewhat longer barrels to get the proper power out of it. But, longer barrel in combination with a silencer tends to shift the balance of the rifle too much forwards.
Second, the bore diameter is obviously smaller with 6,5 mm caliber so this makes a standard contour 6,5 mm barrel considerably heavier than the standard contour 7,62 mm barrel.
Third - I noticed when using short 308 win barrel that the rifle is lighter, shorter and also the overall balance of the rifle with the silencer on is a lot better and there is almost no noticeable forward weight.
Forth - 308 win with it's larger diameter and heavier punch is much more appropriate for use on driven hunts, which are an important part of hunting culture and tradition here. To use the correct type of bullet for the job at hand is very important, but there's no way to go around bullet diameter. Wider is the correct way to go when driven hunting.
I have been reading countless debates about 6,5 mm bullets vs. 308 win for years. So I would dare to say that while both calibers are really excellent and some of the best I ever used I would still say the 308 win is the overall winner and more universal of the two. It tolerates shorter barrels well, punches harder, the bullet is larger diameter which is important especially on driven hunts, and also the rifle is lighter, shorter, handier...