Well, I have a couple of each variety, a Mossberg 500, Mossberg 590, Remington 870 Wingmaster and a Remington 870 Sportsman (a discontinued mid-grade model between the Express and Wingmaster) so I really can't say one is better than the other but my leanings are towards the Mossberg and here's why:
The safety on the Mossberg is ambidextrious that is an advantage for right or left handed shooters or those that practice weak-handed drills as well. There is no problem with working the safety with either hand. Yes, the plastic button is a breakage-prone item (broke one myself years ago) although that's a $10-15 fix with a metal one. Remington advocates rightly point out that the stock button is breakage prone however they do tend to forget Remington's adventures with the J-lock safety button that Mossberg owners can also cite. Again, an easy swap of parts and the J-lock remadies that problem if your 870 came with one.
The slide release on the Mossberg is in a better location for easy manipulation. It falls right under your middle finger where either left or right handed shooters and doesn't require the shooter to break their grip to depress it. How often are you going to need to depress the slide release? Maybe not all that often but for the same reason a lot of us CCW, it's a good option to have even if you never need to use it.
The shell lifter on the Mossberg is skeletonized and rides against the bottom of the bolt when the slide is forward. There is no need to depress the lifter when loading the magazine tube. I've never done it but I've heard of people that have had their thumbs snagged by the lifter of an 870 that is impossible with the Mossberg design. What I have expereinced first hand though was when unloading my 870 Sportsman by depressing the shell latches was having a shell shoot out of the magazine and on top of the lifter with the bolt forward. That was NOT a fun time trying to get the shell out and required taking the gun apart in order to do it. My solution was to dremel a cut about 3/4th of the way along the length of the lifter so if that does ever happen again, a knife blade can be used to push the shell back into the magazine. Current 870s come with the flex tab where all the operator has to do is to pump the action again to clear the stoppage and for older guns, a flex-tab conversion is also available as well. However, the Mossberg doesn't have this problem to begin with so here again I rate that a superior design to the Remington.
Another thing from an armorers persepective is that the Mossberg may have more pieces to keep track of during disassembly, they are easier to repair. When a shell catch goes bad in a Mossberg, simply put in a new one when you reassemble it. If an ejector breaks, it's held in the receiver by a simple screw that even the mechanically declined can replace. When I have to replace a shell catch or ejector in an 870, it requires restaking that part into the receiver. My last command had a few OLD 870s that were 1-2 staking jobs away from being red-tagged as there wasn't enough metal left to do many more parts replacements requiring any more staking!
Where the Remington does have advantages (both real and in theory) are in it's steel receiver and number of accessories avaiable for it. In theory, the steel receiver should be stronger than the alunimum receiver of the Mossberg. In reality though, the bolt of the Mossberg locks into the steel shank of the barrel much like the locking lugs of an M-16 locking into the barrel and being a CATM instructor these days, I seen a lot of OLD M-16s with their alunimum uppers that have fired a lot of rounds, been rebarreled several times and the receiver is still just fine. I think when you look at it from that perspective, the steel receiver superiority of the 870 is only theoretical. What is real though is the sheer number of aftermarket parts available for the Remington. If there's an aftermarket part made for the Mossberg, there is likely a Remington version also produced but if there is an aftermarket product made for the 870, there might be a version made for the Mossberg...but not necessarily. Magazine tubes and barrels are one such example.
Finally as far as the Mossberg being the shotgun adopted by the Military, that's true...partly. When I was in the Marines in the late 80s and early 90s, the Mossberg 590 was the standard shotgun that was phasing out the old 870s and Winchester 1200s we then had in inventory. However, these days as a CATM instructor for the USAF, I have never seen a Mossberg here and have only seen Remington 870s. First were the old wood-stocked Police models that may be the 5-round variety or fitted with the 7-round magazine and bayonet lug. Today they are being phased out in favor of the new 870MCS (Modular Combat Shotgun) using the same 870 design but with Speedfeed-IV pistol grip stocks, variable barrel lengths, magazine capacities, screw-in chokes, rifle sights and a rail on the top of the receiver as well as a barrel/magazine band with a rail section on it as well. To say that the Mossberg is the "official military shotgun" is correct but only up to a point. The 870 is also serving in the military as well.
In either case, you aren't going to go wrong with either design. You just have to recognize the strengths and weakness's of each design and go from there. At the end of the day, it's a Ford/Chevy thing in that both trucks will get you from point A to B equally as well. It just depends on what emblem you want to have on the grill.
My 500 20 gauge doesn't seem to have one at all...