Here are a few of the design differences between the 870 and the 500:
The 500's receiver is aluminum while the 870's is steel. Hence the 500 is lighter while the 870 is stronger (though I've never heard of a 500's receiver cracking or wearing out).
The 870 will accept magazine tube extensions while the 500, due to the way the barrel screws onto the mag tube, won't.
The 870 has its safety at the rear of the trigger guard while the 500 has its safety at the top rear (tang) of the receiver. I personally prefer the placement of the 500's safety but the 870 style is more common.
The 500 has its slide release at the rear of the trigger guard while the 870's is somewhere else (front of the trigger guard)? I find it very convenient to hit the slide release button of the 500 with the thumb of my shooting (right) hand, but that's just a personal preference.