This is a great topic. I absolutely love these shotguns. Here's the big differences in my opinion:
A1 has a metal trigger guard where every base 590 I have seen has a plastic one
A1 has a metal safety selector
A1 will usually be Parkerized while many regular 590s are blued
A1 has a heavier walled barrel
Both have numerous factory configurations but the stock, pump, receiver and action are essentially the same level of "heavy duty"ness if comparing the equivalent 590 and 590A1.
Both have the clean out style mag tube which is an upgrade from the 500.
Both are duty grade, military issued, and will outlast the owner in the case of a normal guy.
Based on use case here would be my recommendation:
If it's for home defense or predator defense - it's common for people to buy the 14in 590 shockwave with raptor grip and then put the Magpul buttstock on. So you get the light weight and tight handling of the 14 inch barrel but still get a proper stock.
If you want large capacity or a fast reload - The 590RM has great ergos and detachable magazines with up to 20 round capacity and because the mag is OEM and only fits in these shotguns we can have the full capacity.
For competitive use - depends on what you are doing. If you're just shooting slugs and moving quick, I'd pick the 590 since it has decent capacity and less weight than the A1. If you need a choke, the A1 model 50774 has interchangeable chokes. Could be worth looking at.
If it's for serious tactical use - The A1 is great. Super heavy duty, and can be somewhat high capacity with the 20in barrel and 9 shot factory tube. In this environment I'd say the extra weight is worthwhile in exchange for the durability.
If its for cool factor/collecting - Pick the one with your favorite look and ergos! Any 500, 590, 590A1 is a quality shotgun that will almost certainly be extremely reliable.