There are only so many stories, so the difference is in the details
Sylvester Stallone was originally cast to play the lead in Beverly Hills Cop. With Sly as the lead, the movie would have been substantially different. And according to Wikipedia (the most reliable source in the world), the movie Stallone imagined for Beverly Hills Cop became Cobra. The details are sketchy, but apparently, Stallone wanted a harder-edged movie than was planned and the part went to Murphy–an easy choice given his popularity on Saturday Night Live and 48 Hrs.
Beverly Hills Cop with Stallone would have been different. For instance, no police cars would have been disabled by sticking a couple of bananas up the tailpipe.
(By the way, the bananas in the tailpipe thing doesn’t work. I tried it.)
The point is, you can change one thing–switching a wisecracking young black man for Rambo, and you have a totally different story, even if the plot is the same.
You can play the game with your own favorite stories. What would happen, for instance, if Judas weren’t an evil, money-grubbing snail of a man, but instead was trying to honor God by arranging a political killing to bring the Jewish nation back to its earthly splendor? What would happen if Harry Potter were set in Brooklyn instead of rural England? How would 50 Shades of Grey be different if Anastasia were a deeply religious woman, trying to resist her attraction to Mr. Grey–you know, aside from the severe reduction in hardware?
Overall, there are only so many stories in the world. Cracked.com, for instance, lists seven successful movies and asserts that they’re rip-offs of other movies. For instance, they say the most recent Star Trek movie is actually Star Wars (actually, what is now known as Star Wars IV: A New Hope). Maybe some of them are. Maybe Star Trek is. But it’s a completely different story, and both movies were enjoyable.
Basically, if you take an existing story, change a few things, and put it in the hands of a different–albeit skilled–storyteller, you have a totally different story.