And you wouldn't consider getting both because......
Really, it's 6 in one hand and half dozen in the other. The Remington is a little heavier unloaded due to the steel receiver but they are both about the same weight loaded because the Mossberg will hold an extra round.
The Remington tends to have a smoother action due in part to the mechanical levering down of the lifter when running the pump back where as the Remington has a lifter that stays in the down position and is a little smoother. However, the solid lifter on earlier models can lead to a nasty malfunction if a shell slips past the shell latches where the shell is on top of the lifter with the bolt forward. Newer 870s have a flex-tab lifter that negates this problem. The Mossberg never had this problem as the lifter rides against the bolt so if a shell does slip past the shell stop, it just falls out.
Remington shooters cite the steel receiver in theory is stronger than the aluminum receiver of the Mossberg but the Mossberg shooter will counter that the Mossberg receiver isn't as prone to rusting as the steel receiver of the Remington. They also cite that the bolt on the Mossberg locks into the barrel (think M-16 here) so the stress on the aluminum receiver isn't a problem.
Remington shooters complain about the breakage prone plastic safety on the Mossberg. Mossberg shooters counter that it's a $15 or so fix with a metal version and then counter with the J-lock safety that in some cases are still floating around on some older 870s.
Remington shooters cite that the Remington doesn't have pieces that can come out during disassembly that can be lost and increase the parts count. Mossberg shooters will counter that the shell catches and ejector are easily replaceable where the staked in catches and ejector of the Remington have a limited number of times these parts can be restaked before there is not enough metal left on the receiver to replace these parts again.
Remington shooters cite the safety location works without breaking your grip if you use a pistol grip stock. Mossberg shooters counter that the safety on the Mossberg is ambidextrous.
Mossberg shooters will cite that the slide release is easier to manipulate. Remington shooters will counter that if kept in cruiser ready condition, it's a moot point especially if you don't have a need to perform select slug drills on a routine basis.
As you can see, both sides have their merits and complaints about the other design and in many cases, both with and without merit. Really it's a matter of personal preference and as you can see, I couldn't really come up with a clear cut winner here so both is truly the better option!