As you are seeng by the replies, two camps with good reasons. My preference is the 870 and I have had mine since I saved all winter to buy it in '74. I have hunted with it, shot trap with it, & when I got a better trap gun, my 870 became my A#1 HD gun & still is. Only one time in the entire time I've had it have I had a failure to extract. I was using some cheap-ass, plastic base trap ammo and the extractor would tear through the base periodically. Some of the hulls were coming out split in multiple places so I suspect something odd about the plastic lot itself.
Do note, early versions of the 870 can jam. There is a mod for that & I converted mine over in '86. It requires a new bolt & tongue. As my brother-in-law was a Huntington Beach dick at the time, I got it at the police armorer's price of $20. What will happen is when the gun is riding in a vertical position such as in a prowl car mount, a hard bump on the suspension of the vehicle could cause a shell to drop out of the tube mag and reside under the bolt & between the tongue. That would make the gun difficult if not impossible to cycle. The new bolt is relieved underneath and the new tongue has a U'shaped slot in it. If you have a tongue without the slot, those can jam.
My 870 has a Choate extended safety, Choate extended mag to 7, Choate blaze orange solid shell follower, Choate pistol grip, & Reminton 18" police barrel. It is a handful even with 2.75" loads.
On the Mossberg, according to the below source, the Mossy 500 is the only shotgun to pass military muster. From the below link -
"Mossberg claims the Model 500 is the only shotgun to pass the US Army's Mil-Spec 3443E test, "a brutal and unforgiving torture test with 3,000 rounds of full power 12 gauge buckshot". The updated 3443G specification requires a metal trigger guard, so only the Model 590A1 variants, which have a heavier barrel and use metal trigger groups instead of the standard Model 500's plastic trigger groups, will fit the requirements. However, the Remington 870 was not submitted to the military for the trials. Remington may have refrained from participating because they did not wish to risk their their already-strong reputation--winning a military contract would have had little impact on Remington's strong police and civilian market, but failing the mil-spec test could have had a serious negative impact on sales."
Mossy 500
At the end of the day, it comes down which one feels the best to you, is priced where you need it to be, Ford vs. Chevy. I don't believe either is a wrong choice.