I just purchased a new FN Police Shotgun (basically a 1300 Defender). As said before, the bolt (of the rotating variation) disengages and pulls the slide back almost immediately after each shot is fired, so it makes for very fast action cycling. However, not everyone likes that. Personal preference mainly, but I enjoy the 1300 action. I also prefer the safety in the front of the trigger guard opposed to behind it. I feel the slide release is in a fair place on both the Remington and the Winchester.
If I recall correctly, here are some other differences:
1) 870 engages the bolt in the hood of the barrel whereas the 1300 engages into the barrel with four lugs. (Correction noted, thanks Dano523 and brentwal)
2) 1300 requires 1 pin be pulled to remove the internals, 870 has 2.
3) 870 has a steel reciever, 1300 has aluminum.
4) As also mentioned before, the 870 has a massive aftermarket following, however, the 1300 isn't too far behind when it comes to add-on toys as far as I've seen.