None.
and why do many people go w. 300 over 30.06?
Cuz eet's a maaugnum, by gawd!
.260 is the perfect whitetail/mulie/ sheep / goat chambering, even for looong range. But .243, 7mm-08, and a few others are up there too, as nearly-equally-perfect. .257 Robts too of course, but then you're in a long action.
.30-'06 though is (as is often said) never a mistake - you've just got so many loadings to choose from, including some really cheap ones, and including 150 grainer loads, perfect for deer, and even the reduced recoil 125 gr loads, also perfect for deer. So many good reasons to choose .30-'06 over .300 win mag - cheaper ammo being the main one, but also less recoil, less muzzle blast/noise, longer brass life, MUCH more selection in factory ammo (as mentioned), and good ol' American nostalgia. And you give up, in essence, nothing. Sure, it's 150 fps or so slower, but that translates to what, about 3/4ths" less drop at 300 with a 200 yard zero? Long shots like that are exceeding rare (*especially* in the East), and if they are taken, is 3/4" gonna make the difference? Possibly, but seriously unlikely.
Although I like .260 and .280 better myself, nothing more American than .30'-06. My favored big game hunting chamberings where a long shot (past 150 yards) might be needed are, roughly in order: .280 rem, .260 rem, .243 win, .280 remAI, .30-'06, 6.5x55, 7mm-08, 7mm RSAUM, 7mm WSM, and for really big stuff, .375 HH mag - if that helps you get a feel. Oh, and of course .270 is fantastic, but I don't do .270 anymore. Not a big fan of .308, but it's a very good all around choice too, of course.
In my view, if you need to go bigger than a .30-'06 (for say, coastal brownies, polar bear, bison, etc., if you get up north), then forget the .300 win mag and get one of these: .338-'06 A-square, .35 Whelen (or Brown-Whelen), 9.3x62mm, or .375 HH mag. If you need more RANGE than .30-'06, go with a 7mm like 7mm WSM or .280 Rem AI, or even a barrel-burning canyon/beanfield-crosser in a sub-7mm-caliber, like .270 WSM, 6.5-.284 Norma, 6.5mm-'06 A-square, or the new .26 Nosler.
I'm not a HUGE hunter who's killed dozens upon dozens of heads of game, but I hope to be before it's over with....but this is what I think I've learned so far.... Recoil is your enemy (flinch, etc.), and so is expense. Smaller is usually better, but not always. Accuracy is almost everything. Bullet construction is next. Chambering is a personal choice, but overall, not to terribly important, so get what floats your boat (if you can afford it and handle the recoil).