Having owned several of both I feel slightly qualified to answer this.
The 870 is able to take more abuse and neglect. It will run when the BPS won't. This is not a matter of quality exactly, but rather the internals of the BPS are much more tightly fitted. There is little room for dirt and grime in the BPS. This is also why the BPS action is very smooth right out of the box. The 870 has more room for gunk and garbage in the action before it is taken out of commision. The 870 action will also get very smooth after 10,000 rounds or so.
The BPS safety is right where it belongs, on the tang. The 870 safety is the abberation we have been conned into believing is standard. This is purely personal opinion, but one I feel strongly about.
Both will take more rounds than you and two others could put through it in a lifetime. I would guess-timate that the service life of both is well over 250,000 rounds of handicap trap type loads, and more of lighter loads. You may break a spring or something here and there, but the guns will be solid.
The loading of the BPS is a real PITA when target shooting, or shooting any volume of shells. You must have the slide closed, and load the rounds into the magazine, then depress the bolt unlock lever to cycle the action if the chamber is empty as it usually is in target shooting. Hunting this is really not a big deal, as you will be loading the magazine anyway, and loading a lot less shells. With the 870 you can drop one in the ejection port and close the action before loading into the magazine. MUCH easier in target work or field work when starting with an empty chamber. Topping off the magazines is the same for both.
The BPS is very well made, and a great shotgun. The 870 is a great gun too, and works well.
If I could only have one or the other I would pick the 870, and mine would be the express so I could buy two for the cost of one BPS.
If you have any specific questions let me know. Hope this helps.