I don't disagree with most of the posts, but I shoot both, and I don't agree that recoil is a huge issue unless your playing PRS, tactical steel, or another rapid shot, recoil management game, in which case even .308 is way too much without recoil mitigation.
My .300WM is a heavy rifle with a full fluted plama barrel. With either the Badger Brake or an EA can it has considerably less felt recoil than a .308 without recoil mitigation. It's pleasant to shoot an 80-100 round match. I don't feel like I've done anything special when I'm done. Yes, the scope jumps off target when you pull the trigger, but you're not feeling it shot to shot, and that's with 210gr running around 2,850fps.
Obviously it's way more than a 6mm or 6.5mm with a big brake or can, but that's virtually no recoil. Those are also have very little energy at longer range, and they are much more affected by poor, windy conditions compared to a 210gr out of a .300WM. They're great for paper or steel, or game at medium range, but not good for taking longer shots ethically IMO.
I shoot sporting clays frequently and if I shoot a round and then a couple of boxes for break n' chip, or team 5 stand I have a visibly bruised shoulder for a couple of days afterwards. I never start noticing till after more than 100 birds, so I guess I'm not very recoil sensitive. I'm used to high round count 12ga, and long matches.
I've certainly shot many .300WM rifles that would beat the hell out of you, but there are plenty of ways around that if what you're after is a lot of energy and penetration out in teeny tinny land. It's not "THE BEST LR" cartridge for sure, but it's certainly the most available, affordable to shoot, and I don't find it difficult to tune it to the rifle.
I shoot my .300 WM a lot less than my lighter and faster rifles, but then I send way more rounds down range competing in tactical steel rather than hunting. I know that .300WM has fallen out of favor, but I'm still a huge fan, know lots of hunters and LR hunters who love it, and don't see that changing. There certain games where you still see .300WM in the winners circle frequently.