You're welcome, asp. A coupla things....
There's no apples to apples comparison possible between the 870 and AR series. But, a few things can be compared.
Range,the AR wins this by a couple hundred yards.
Effect, with center mass hits, both are quite effective.
Overpenetration, both will, so be sure of the background before popping caps.
Cost, here the 870's a clear winner. One can have a stable full of special purpose 870s for the price of a bare bones AR. Ammo, depending on what you buy, is not terribly different on a round to round basis.
"Political Correctness" and other such #$%^*, AR type weapons are more scary to the sheeple. This applies to court, also. Shoot a perp with an AR, there's too big a chance they'll use the looks to inflame the jury, who will NOT include anyone with either a shred of common sense or weapon experience. If you're gonna use a rifle, use your grandad's Model 94 "Deer" rifle.
Ease of use, while both require practice, the 5.56 mouse round and it's carrier, the AR, are easier on rookies that can be brought up to speed quickly. Back when I trained COs, it was rare that a rookie flunked the Mini 14 portion of training, while many had severe trouble dealing with the 870's kick. I heard one rookie refer to the Mini as a "Candy gun". Apt...
Versatility, the shotgun wins this one by a mile. Anyone who ever has eaten venison taken with the .223 knows how much less edible meat there is on a deer shoulder post shot. More so for small game, woodchuck tend to fragment.
And, it's harder to maintain rifle skills than shotgun skills. Rifle ranges are less common than trap and skeet ranges. and while some poo-poo the clay games as not relevant for "Serious" work, they do teach familiarity and improve accuracy. Anyone doubting this is welcome to shoot some trap or skeet doubles. In trap doubles, one has about 3 1/2 seconds to destroy two 4" clay targets moving in different directions at 35-50 MPH.Best I've ever done is 39/50, and I'm fairtomiddlin' with a shotgun.