What's the Purpose of Such a Gun when there are already so many to chose from for say, protection? Serious question.
By "such a gun," I'm assuming you're speaking of civilian AR-15's, civvie AK lookalikes, mini-14's with aftermarket stocks, and such? (Just making sure I'm understanding you, and that you're not talking about the Anschutz target rifle above, for example.)
Most people, including me, who own small-caliber self-loadering carbines own them largely because they are fun as recreational target rifles (the AR-15 platform is probably the most popular centerfire target rifle in America), and a lot of us Gen-X and Gen-Y types are also drawn to the more modern aesthetics. They don't beat up your shoulder like full-power rifles can, and ammunition is cheaper, but they have much better ballistics than .22 rimfire rifles.
AR-15's in particularly have a reputation for very good accuracy; for example, Rock River Arms guarantees that its carbines will shoot a 1" group at 100 yards. AK lookalikes are a bit less accurate (more like a Winchester 94 in .30-30, say 3-5 inch groups), but still fun guns to shoot.
But to address your specific question, as far as home-defense use goes, IMO a .223 carbine like a mini-14 or a 16" barreled AR-15 type rifle has advantages over both handguns and shotguns.
Versus handguns, advantages include:
--Easier to shoot well under stress than a full-power handgun, because of the longer sight radius and the fact that your hands are further apart.
--WAY less recoil than a full-power handgun of comparable effectiveness. For example, full-house .357 loads can be pretty brutal to shoot, but a .223 carbine offers similar effectiveness in a firearm that recoils like a .22 (because it
is a .22; a .223 bullet weighs about 1/2 to 1/4 as much as a typical handgun bullet, and is pretty close to the bullet weight of a .22LR).
--.223 JHP's are less likely to overpenetrate in building materials than most handgun rounds.
--Might possibly have a greater deterrent effect against an intruder than a handgun (don't know how real that effect would be, but a carbine definitely gets your attention than a small pistol)
Disadvantages of the carbine vs. a handgun:
--Lack of portability. Thanks to the shoulder stock, thicker and longer receiver, bulkier magazine (rifle rounds are longer than handgun rounds), and the minimum 16" barrel length and 26" overall length required by Federal law, a carbine is way less portable than a handgun. In and around your house, the carbine is great, but you can't exactly carry one on your person for self-defense, and they are definitely harder to travel with. Even with the stock folded for storage, my mini-14 is around 30 inches long, 8 to 10 inches high, and weighs 6 or 7 pounds. Compare that to the size of a handgun, and think about the transportability/concealability implications:
--A safe big enough to hold a carbine is more expensive than a handgun safe, and you can't stick one just anywhere in your house. So a handgun is easier to store than a carbine, although if you have a safe then it's a moot point.
--A handgun is more easily kept in a higher state of readiness. A 9mm pistol can be safely kept with the chamber loaded, safety on, as long as it is secured from children and unauthorized users. Rifles are generally stored chamber empty, magazine loaded, like a shotgun. Meaning it takes a second, and two hands, to get a rifle going. Generally not a big deal, but it's a difference.
--You can hold a carbine one-handed for a minute to open a door, help your kid to safety, or whatever with your other hand, but carbines are awfully hard to operate one-handed, unlike a handgun.
--Rifles are somewhat louder than handguns, which may slightly increase your risk of some hearing damage if you ever have to use one in a defensive situation. Given the rarity and the gravity of such an event, that's probably not an issue.
--If you live in the city, you may have to drive a bit further to find a rifle range. Many indoor handgun ranges don't allow centerfire rifles because of backstop erosion issues.
Advantages of a small-caliber carbine vs. a shotgun:
--The carbine doesn't have the brutal recoil of a shotgun (a .223-caliber bullet is a HECK of a lot lighter than a .729 caliber shotgun slug or a heavy buckshot load); you certainly lose some incapacitation ability vs. a shotgun, but you gain precision and the ability to more easily recover from a miss.
--Magazine capacity. A typical defensive shotgun has about the same capacity as a revolver, six to eight rounds. A typical carbine will have a magazine capacity of 20 or 30.
--Ease of loading. If you choose to store the gun completely unloaded, it's a lot easier to insert a magazine in a carbine than to feed six or eight shotgun shells through a loading port.
--The carbine offers less chance of stray projectiles than a shotgun with buckshot at anything more than across-the-room distances, and penetrates much less in building materials than a shotgun with slugs.
--A carbine is more fun to practice with than a shotgun or a full-power rifle. Target shooting is more fun with a rifle that doesn't kick much; honestly you'll probably never fire a gun in self-defense, but target shooting is an end in itself. Shooting a couple hundred rounds of .223 won't even leave your shoulder sore, but even fifty rounds of 12-gauge will beat you up unless you're wearing a shooting jacket. Also, carbines are much cheaper to shoot than shotguns,
--Carbines are much less delicate than shotguns, and tend to be more reliable. Partly because shotgun barrels are so freaking BIG, they tend to be quite thin in order to save weight, and they also operate at much lower chamber pressures. So it's easy to dent/bend the barrel of a shotgun. Shotgun receivers and action components also tend to be much more delicate than rifle parts.
Disadvantages of a carbine relative to a shotgun:
--Somewhat less effective than a shotgun. Obviously, a .223 caliber carbine will have less incapacitation ability than a .729 caliber shotgun slug, or the eight or nine .355 caliber projectiles in a 000 buckshot shell.
--Carbines generally cost more than shotguns.
--No carbine can probably match the deterrent effect of a .729 caliber shotgun barrel pointed at the guy breaking in...
Advantages of an AR-type carbine/civilian AK lookalike/mini-14 with aftermarket stock over a more traditional rifle for defensive purposes:
--WAY less recoil than a full-power rifle.
--WAY less penetration in building materials than a full-power rifle.
--Lighter than full-power rifles, even ones with 16" barrels.
--Smaller rounds = smaller, lighter magazines for a given capacity.
--Pistol grip stock allows you to use a low-ready position (gun pointed at the floor in front of your feet) instead of the port-arms or high-ready positions that a traditional stock would lead you to use. Low ready is safer in confined spaces, IMHO.
--Pistol grip stock makes it harder for an attacker to lever the gun away from you by grabbing the barrel.
--With a pistol-grip stock, it is easier to short-stock the rifle (put the butt of the rifle behind your shoulder) while you're moving through a doorway. You can do this with a traditional stock, but it bends your wrist like a chicken wing and would make it very easy to have the gun taken away from you.
--AR-15's and civvie AK lookalikes are more robust, durable, and reliable than some traditional designs (coughRemington7400cough).
--Easier to mount a flashlight on an AR (IMO a safety feature).
--.223 and 7.62x39mm are much less expensive than full-power rifle calibers.
Disadvantages vs. traditional rifles:
--The carbine will have WAY less energy at long ranges (more like a .30-30). For defensive purposes, this is not a big deal, as you can't exactly claim self-defense if you're shooting at somebody 500 yards away...
Someday, when I can afford to, I plan to get one of these, but I'll have to sell the Ruger to swing it:
Rock River Arms carbine with the mid-length gas system. All the advantages of the mini-14 as far as light recoil and reliability go, but it is more ergonomic (safety, mag release, and bolt stop are more intelligently designed), looks great, excellent accuracy, high-quality sights, and built-in optics rails. The only downside is price (MSRP is $1100, eek!). But IMHO that's the perfect target-shooting gun that can also serve as an excellent defensive carbine.