As stated above, most use 160gr bullets in the 7mm Rem and 180gr bullets in the 300 Win Mag.
The 7mm will produce less felt recoil in rifles of the same weight as the 300, by as also stated above recoil is subjective and no two people experience it in the same way.
At the end of the day, less felt recoil by the shooter will generally produce better shooting, and promote extra quality practice, which is always a good thing.
(I too prefer the 300 WSM over the 300 Win Mag due to the improved efficiency of the short, fat powder column requiring 10% less powder to produce the same velocity, which equates to less recoil). I shoot my WSM better than shot my 300 Win Mag when I had it...that rifle's felt recoil was sharp compared to my rifle in WSM.
On game performance, the 300 will provide a slight edge in impact due to the slightly larger frontal area of the bullet. The 7mm has a better BC and will shoot just a bit flatter...but @ 500 yards, most hunters from field shooting positions won't be able to realize the difference. (not referring to experienced LR competition shooters here with LR set ups and higher powered tactical scopes, but rather the average hunter with regular hunting rifles and scopes))
A 7mm Rem Mag shooting a 160 gr AccuBond @ 3046 fps with 3296 ft lbs of energy will still be doing 2209 fps @ 500 yards with 1733 ft lbs of energy and be down 35.7" (with a 200 yard zero).
A 300 WSM shooting 180 gr AccuBond @ 2978 fps with 3545 ft lbs of energy will still be doing 2117 fps @ 500 yards with 1791 ft lbs of energy and be down 38.2" (again with a 200 yard zero).
Both have sufficient velocity for reliable bullet expansion at that range and with proper bullet placement, will cleanly kill an elk or moose. Neither animal is going to be able to tell the difference between the two.
The slight edge in penetration goes to the 7mm as this bullet has a Sectional Density of .283 vs .271 for the 30 cal bullet...but again, probably not significant enough to tell a difference on game unless heavy bone encountered...but with both, stay off that heavy shoulder bone...the minimum bullet that will reliable penetrate an elk's heavy, dense shoulder bone and still make it into the vitals is the .338 cal (338 Win Mag) 250 gr bullet with an SD of over .300.
I have used the 7mm Rem Mag and 300 Mags (Win and WSM) on numerous moose and elk over the years and have found both to work well. Many get caught up with thinking about longer range shooting 400-500 yards (or more). In 37 years of hunting these animals, my average shot distance is just 137 yards; with my longest shot on a moose to date being 220 yards, and an elk at 475 yards (where I severely underestimated the range as I watched the bull approach from 1/2 mile away through my scope cranked up to 10 power - I did hit him with the first shot, but just across the brisket below the vitals).
Myself, I prefer the 7mms over the 300, and if needed, I usually jump up to the 338 for more power. I do like my 300 WSM (my first custom rifle) that will consistently produce 1/2" groups with its preferred load of 180 gr AccuBonds, but tend to grab my 7 more often when heading out. (Although I now have a LH 7MM STW that replaced my RH 7mm Rem Mag).
At the end of the day, buy the rifle and scope combination that fits you, in the cartridge that you are most comfortable in shooting, and enjoy! (Both the shooting and hunting adventure, and the fruits of your labour!)