Talk to the armorers servicing both and they will all tell you the Mossberg falls well short of the 870 in reliability.
That was my experience when working as the armorer of an agency with 160 sworn officers. We issued a variety of shotguns (they were all pumps that were confiscated by a large urban police department...we would cut the barrels to 20" and issue them to our officers...it really hurt to cut off the barrel of a nice Model 12 with a factory rib) but probably 40% were Mossberg 500's and 40% were 870's. This was about 20 years ago.
The Mossbergs were a constant headache. The major problems were safety buttons that would simply fall off when the screw that holds them in place would loosen during normal handling, plastic safety buttons that would break into two pieces when you would tighten the screw to try to keep them from falling off, weak magazine springs that resulted in misfeeds and shell stops that would bind and cause jamming. There were timing issues with shell carriers and loose and cracked wooden buttstocks as well. I replaced many cracked wood buttstocks with synthetic ones and then they held up fine.
The 870's were trouble-free with no issues that I can recall...might have been a trigger group that needed replacing on one gun, but can't remember for sure. The Ithaca 37's and Winchester 1200/1300's also worked well. There were a few off-brand shotguns such as Nobles that were awful and made the Mossbergs look like diamonds in comparison.
I have seen a few problems with 870's in later years at a police academy range, mostly shell stop malfunctions, but they are still rare issues.
Admittedly these guns (including the 870's) were used (and abused) harder by our officers than a typical hunting gun would normally be. We got rid of the Mossbergs when the budget allowed. After I retired the department bought Remington 11-87 semiautos and had problems with them, so they traded for new 870's and have been satisfied since then.
Maybe the newer Mossbergs are better now. I know lots of people like them and a typical hunting shotgun doesn't take the abuse of a law enforcement weapon.
