My first thought is to the AR-15 platform over the shotgun:
Ammunition: Simply put, the 5.56 or .223 will give you more shots per pound of weight. If this is a bugout situation (or even a regular travel scenario) you will have less than 12 rounds per pound of 12 gauge shotgun ammo. The whole point of the AR design was to carry more rounds than the .308 rifles it replaced (M-14 and M1 Garand). Second, the 5.56 is a very good self-defense ammunition against unarmored targets. The properties of the round mean that it is less likely to over penetrate the target, as you would get with .30 caliber rounds (7.62x39 being among them).
Range: The 5.56 mm, while not be a great long-range round, still has the ability to reach out well beyond pistol and shot gun ranges and hit targets accurately. If you think you may use your rifle to take small to medium sized game this range will give you a distinct advantage over a pistol or shotgun.
Capacity: A shotgun will give you something like 6-10 rounds unless you go for something expensive like a Saiga 12 gauge AK chassis with a removable drum magazine. A standard AR-15 has removable 30-round magazines. If you are in a defensive situation, that means more rounds ready to go in a lighter, easier to handle platform. I like the Saiga shotguns, but they are much heavier than an AR and often are too much gun for some shooters.
Accuracy: The AR-15 can be built into a highly accurate firearm, not so much with a shotgun. Yes, it is possible to miss with a shotgun, despite what people say. You don't get the precision of a good rifle, the range, the light weight, but you can still miss.
Maintenance: The AR-15 is a bit fiddly with small parts compared to it's AK-47 counterpart (sorry, I know you said don't get into this, but it's important) and the tolerances are much tighter, making it more prone to fouling due to dirt and residue; as such, it often gets maligned for being a difficult weapon to maintain. One of the reasons for this is that the lower-end AR-15s are not particularly well built, your run of the mill WalMart Bushmaster simply will not run as well as a $1,600 battle rifle. These low-end ARs are good for going out to shoot cans, but you don't want to trust your life to one. My AR-15 is an All Star Tactical lower receiver with a PWS 14" upper with a 2" pinned muzzle device. I opted to get a piston-driven upper instead of the usual (and often cheaper) direct gas impingement. This means my AR-15 runs colder and cleaner than your typical AR because it has an AK-47 style piston running up the gas tube to the gas block; this keeps the dirty gasses from each shot from blowing back into the receiver where they will eventually foul up the action. Simply put, I can run my AR-15 longer before needing to clean it.
The piston driven uppers do add a little weight to the rifle, but it still comes in well under a 12 gauge shotgun.
Now, this is all comparing the two firearms originally asked about. Depending on where you are going and what you may encounter, you may also want to consider a Henry .45-70 guide gun. This is a not-screwing-around bear gun that packs a massive punch at short range. The .45-70 was originally a US military cavalry round and it hits like a hammer but has the ballistics of a mortar round - the round is so heavy it just can't get very far. However, it should bring down large game such as elk or moose with one good shot.
Another option for a bugout is to go smaller. A Ruger 10-22 Takedown breaks down into two parts and stows easily inside a backpack. The .22LR ammunition is light weight and probably gives you the most rounds per pound. This rifle, and others like it, are favored by bush pilots for their ability to take small game like rabbits in a survival situation. If you were going to keep a rifle in with your bugout bag, this would be a good option.
