IMHO, and my opinion is based strictly on my three BPS and 31+ years of 870 use at home and on duty, the similarities are:
Both are all-steel. The BPS has a slightly larger receiver and no second cut on the right for the ejection port, so the BPS weighs a few ounces more than a comparable gauge 870. Weight on both sits pretty much between the hands.
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LOP on both this 20 ga Ducks Unlimited 870 and 16 ga BPS Hunter are the same. The BPS has a slightly wider trigger than the 870, which I like. Outside of the wider trigger face, the trigger action on both guns feels similar when I fire.
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Both guns are powered by dual action bars that make the forends pretty stable. Reach to both forends is similar, but the BPS has less forend area to grab on to due to the slightly smaller forend and shell ejection cutout on the bottom. To me its a push, but smaller shooters could possibly hamper ejection of the BPS if the hand or wrist is back over the cutout when racking the gun.
The forend release on the 870 is a little bit easier to use than the BPS release, which is sort of tucked in closer to the receiver.
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To me, the BPS has a bit less forend rattle than an 870. It may be because the magazine tube is a touch larger and shells drop through the cutout in the back of the forend, so maybe the side to side play is less to keep them from being hung up? (Just a guess.) Barrel removal is identical, unscrew the knurled cap on the magazine tube and slide the barrel out of the receiver.
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The sighting plane on the vent rib guns is pretty close, but the BPS rib sits a little taller than the 870 rib. I prefer the look of the 870 profile. I like how the receiver is smooth and blends into the rib while the BPS has a dorsal ridge on the receiver that contours into the taller barrel rib. The newest 870’s are drilled-tapped for a rail or direct-mounting sights. (Bottom are two 20 ga, new Rem Arms 870 on the right.)
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The rib of the 870 has rib risers that are closer together and the rib is thicker and closer to the barrel. In high volume shooting I would think the BPS would keep mirage down a bit more. But I highly doubt I will be shooting in such volume, so for me this is just a cosmetic difference I guess.
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For left handed shooters, the BPS is ready to go right out of the box. The top-mounted safety means no trigger guard mounted safety to play with, no ejecting shells zinging right past your nose, etc. A downfall to the BPS single-port receiver is fast reloads (“Combat loading”) are slower, swapping loads and unloading is a bit more of a hassle, and checking to confirm the chamber is empty is different. In addition, there are scads of factory and aftermarket parts and upgrades available for the zillions of 870’s out there, while the BPS has a much, much more limited selection.
We all know Rem Arms is the latest in the Remington saga, but luckily just about anyone can work on an 870. The BPS is not being made anymore, and should something fail it may be a bit harder to get something fixed.
So, for HD duties, I greatly prefer the 870. In the field, i like the BPS a bit more. YMMV.
Stay safe.