I thought knowing this stuff was a prerequisite for Yukon residency!
But seeing as how you live in a place where the answer might have serious implications for you, I'll do my best to give you a serious answer.
There are pros and cons to the choice of either firearm, and the simple answer is to choose the one you shoot best, but this poses more questions than it answers, so lets look at each.
A shotgun should be viewed as a powerful short range rifle, and when loaded with a good quality slug like a Brenneke, it is more than adequate for killing a big bear. It is the sub-adults that are the trouble makers though. The proper shotgun for bear defense is a 12 ga with a 14"-20" smooth bore barrel, a repeating action, and a cartridge capacity of not less than 5 rounds. It should be equipped with rifle sights and a sling. Beyond that hang whatever you like on it (sidesaddle , flashlight etc.) but try to keep the bulk to a minimum. A suitable shotgun is normally less expensive than a suitable rifle, but it might prove to be more prone to damage under hard use, as the barrel and magazine are thinner than a rifle barrel, and a rifle tends to be less bulky. Thread in chokes will help protect the muzzle, and a ventilated hand-guard will help protect the barrel from dents if the gun is dropped or you fall on it. Where the shotgun really shines is the selection of ammunition that is available to solve a bear problem. It ranges from scare cartridges to less lethal rubber buckshot and rubber slugs right up to lethal full powered shot and slug loads. The selection might be too good, in that it is possible to have the wrong ammo in the gun and you end up killing a bear you meant to scare or not scaring a bear you meant to kill. Shotgun shells tend to corrode faster than rifle cartridges, and should be swapped out of you gun once a month or so if you carry it every day.
A powerful rifle I believe is a better choice if you are concerned about bear defense in the back country, or if you are responsible for the safety of a group of people which might pose a problem of distance as it is human nature for people to string themselves out. But also the rifle bullet tends to produce a more severe wound channel and bullet penetration tends to be straighter and deeper than a shotgun slug. But unlike the shotgun, the rifle only does one thing well, which is a lethal response to a bear problem. Others have disagreed, but in my experience, some bears do not respond well to live fire as a scare technique, in fact I've shot at bears at close range with a rifle and a handgun only to have the bear approach me; sometimes they'll run off, but most often the grudgingly give way and slowly retreat in a stiff legged gate. Just to be clear, these are white bears I'm talking about, so your milage may vary with brown ones. But as a rule a grizzly has a larger personal space than a polar bear, so its easier to get into trouble with him.
If you are in a populated area a shotgun doesn't pose the over penetration problem that is associated with a powerful rifle, particularly if you are thinking about the .375 class of rifles. If you need to shoot a bear in a town or crowded camp, that is the one scenario where the use of buckshot makes sense to me, otherwise your gun should be loaded with slugs. When you use scare cartridges, single load them, then if you need a fast lethal shot, its right there in the magazine and all you need to do is work the action rather than attempting to feed more rounds into the gun. I prefer not to use scare cartridges in guns loaded with lethal ammo. Cracker shells in particular can leave partial obstructions in the barrel. Having said that I've never seen a shotgun come apart from firing a slug after a scare cartridge, but I believe the possibility exists, particularly if a cracker shell explodes in the barrel as they occasionally do, and is followed up with a slug. I believe that a starter pistol for scare cartridges is a better way to go, or if there are two of you, one shotgun could be a dedicated scare gun and the other a lethal gun.
Your bear gun should be equipped with a good set of irons that you can shoot with, even if you have a low powered variable or fixed power scope, or one of the new-fangled electronic sights. Don't put a 10X scope on your bear gun. Practice on short range targets and if you can arrange it, on moving targets. You should strive to make good repeat center hits in the minimum amount of time without taking the gun off your shoulder.
The main premise of bear defense is that is the bear can't touch you, it can't hurt you. Everything you do is towards preventing the bear from making contact with you. Shooting a bear at 50 yards is not self defense. Scaring a bear away is the preferable action, but should you have to shoot for blood, the purpose of shooting is to stop the bear's forward progression. A chest shot won't do it, so your target, if not the chest, then is the head, the spine, or a large supporting bone. The brain is only as wide as the snout, so don't be fooled by that wide massive head. If you break a big bone, it is to give you time to quickly follow up with a killing shot.