Hi there,
It's a tough call. I just had to make the same decision myself.
Ultimately, I went with the Mossberg 590A1 Mariner with a Vang Comped barrel. You can buy these directly from Vang Comp Systems for about $545.
I just took mine through a Tactical Shotgun Course, and it performed better than I could have ever hoped.
I think the placement of the safety is certainly an important issue. Even if you're right handed, you will sometimes have to transition to your weak side.
Another difference between the two is that with the slide locked forward, the elevator in the Mossberg automatically goes up and leaves the mag tube unobstructed. The Remington's elevator will stay down unless it's pushed and held up. There are pros and cons to both systems, and it's a really minor, minor detail. But with the Mossberg system, there's no elevator to push through when loading and especially when you're unloading tactically (by which I mean depressing the catch to release the shell from the tube.)
To be perfectly honest, the biggest thing that swung my decision was simply that a person whose opinion I really trust carries a Mossberg. Sometimes it just boils down to that...
By the way, I can't stress enough how fantastic the Vang Comp barrel is. If you're not familiar with it, it's a barrel modification that includes porting and back-boring. What that really means for the shooter is a huge reduction in recoil, faster follow-up shots because of reduced muzzle climb, and an incredibly tighter pattern over a standard cylinder bore.
We spent some time patterning our guns on the first day of class and my Vang Mossberg shoots a 4.5" spread at 16.6 yards with Federal Tactical 00 buckshot! Every other shotgun was between 10-14", which is about what you'd expect. That simply means that at 25 yards I could still probably put all 9 pellets into a headshot.
I'm not affiliated with Vang Systems, but let's just say I've become a true believer

I'd rather give up my sidesaddle and sling before I give up my Vang barrel...
Hope that helps...