TEB,
I don't know how much experience you have with shotguns so far. Both your announced options are good solid guns, even though Winchester has indeed closed its doors, FNH (Winchester's parent company) is still producing the 1300- it's just marked now as the FN Police Shotgun. You can see it at
http://www.fnhusa.com/contents/sg_police.htm . I have one, mine at less than $300 total was IMO a great bargain for all the features it offers. I happen to prefer 870s by a wide margin to all other pumpguns I've ever owned/handled/shot, but that's just me. You should make up your own mind based on experience.
I'd advise you to handle and if possible shoot examples of both guns, and see which one fits you better and which one you like more. The two guns have a lot in common but are very different in control layout and in some degree in function as well (for example, the FN/1300 has 'speed pump' which means the action is unlocked the instant the hammer falls, the Remington depends on a slight forward bump of the forearm in recoil to unlock the action so it can be pumped). Some folks like the Winchester/FN method, others like the Remington. And so on...
The FN/Winchester has an aluminum receiver and a rotating bolt, that means the receiver is lighter than the Remington but it is a bit longer as well, by about an inch or so. That of course means the overall weight of the gun is lower for the FN/Winchester than for the steel-receiver Remington, and the overall length of the FN/Winchester is a bit longer than a Remington with a comparable length barrel and stock. And so on...
Both guns are good reliable designs, both will most likely still be shootable when you are taking the long dirt nap. It is a matter of personal preference, I have all three (multiple 870s, two Winchester 1300s and one FN Police Shotgun) and would not feel underequipped no matter which one was available when the dance opened. Handle them both, shoot them if possible (friends, or nearby gun clubs with rentals available might help there), and then make up your mind based on your own preferences and experiences.
Good luck, and stay safe,
lpl/nc