I have no experience with the 06 but I love the .260 Rem. I shoot an HS Precision 260 Rem. It's my go to rifle for popping gophers, yotes, paper, and Mule deer. I prefer .257 1/4 bore for white tail. Have some info I'd like to share.
.260 Match Ammo Comparison Test
Notice the 1000 meter comparison to the .308. I'm just surprised the 260 R hasn't caught on as an all-around-one-gun of choice. I would not hesitate to hunt moose or elk with it though I do go to my 300 win for them simply because I am not a one gun guy.
Besides a don't have to re-size brass to reload and factory ammo is available.