The 300 magnum gives you about 600 ft/lbs more energy, and about 50 yards more useable range, but there is also an increase in recoil to the tune of about 35-40%. More power is always a trade of for either a heavier rifle, or more recoil.
Personally, I have used the '06 on some large moos, and never found it to be lacking. I have also used the 300 magnum, and it is a fine cartridge as well. Had I to choose one, as I grow older, it would be the '06. With a 168 TSX or a 180 Partition at good velocity, game just flat lays down and dies.
I also have a 45-70 and a 405. The 405 in a 9 pound rifle still comes back pretty good. Ten rounds off the bench is enough to get your attention. My wife killed her first moose with a 243, and it did the job with a 95 gr X at 40 yards, but that is a use what you got gun, not one I would buy for moose.
My vote is for the 30-06 with good bullets and lots of practice for shot placement.