A heat sheild wont fit a 590A1. At least thats what it says on the mossberg website. Of course, you could always swap the safety and triggerguard on a standard 590 with a heat sheild, or put a standard barrel on a 590A1...
Standard 590s are available blued, parkerized, or marinecoate. Of course not all are available in all finishes, for example the only blued one is 20" with a heat shield and the only marinecoat is a 20" without it. Both have synthectic stocks and bead sights. I cant be sure of the rest of the configurations without looking at the website, other than the fact they're parkerized and there are plenty of options.
Anyway, back to the original question, about the only advantage I am aware of is the magazine tube, and one more round with the 20" models compared to the 20" 500s (both in 18.5" are 5+1) . I'm not sure if the 590A1s have it, but my 590 had a piece pressed into the end of the magazine tube to retain the spring when the cap was removed. It came out pretty easy though.
One disadvantage to the 20" 590 is you have to file the bayonet lug off if you add a surefire forend - I think. Not an issue for me because I'd be more likely to do a hipshot with my pistol if someone is close enough for a bayonet (my pistol is the first thing I pick up).
I prefer the 590, mainly due to the swab-through magazine tube and one extra round. Well, three compared to the 500 I used to have.
I've edited this post so many times I think I've finally got it right...