Lets start this by saying that if you think story and plot are the same thing, you're one of hundreds, if not thousands of writers out there making the error... it's okay and actually, it's pretty easy to fix with just a few simple recalibrations in our thinking.
First of all, we can all be forgiven for assuming they're the same because frankly a lot of movies we go to watch are making the same errors. Storytelling has become very VFX led, spectacle led (as it did shortly before the New Hollywood changed it up) and budgets have become inflated. I argue this is because the makers are assuming that audiences are wowed by bigger, better, faster, louder when in fact, audiences are wowed by stories. Why does one well placed explosion, featuring a character you care about, create more emotional attachment than four hundred beautifully choreographed ones in an aerial wide shot?
By getting back to what makes a good story, we can help usher in a second wave New Hollywood, where budgets come down and creativity goes up. But I digress. Why is story different to plot? Let's break it down...
1. Simply put, story is the internal journey of the character, it tracks them from where they start up until where we leave them at the end. Usually, (and rightly) this character starts off with some kind of problem and through the course of the story, corrects that problem and learns something about themselves in the process. The story is INTERNAL to the character and specific to that character. To better explain it, let's take Jurassic Park. This film is not a story about marauding dinosaurs eating people (thats plot but we'll come back to it in a moment) this is a film about a man who vehemently does not want kids, nor does he like kids, learning that he does in fact possess the ability not only to like kids but to respect them. THAT is the STORY of Jurassic Park. It's not an accident he gets stuck in the park with the kids guys, sorry.
2. So the plot then is EXTERNAL. The plot is usually derived from decisions the character makes (when done right) but can also be external pressures upon the character that create the story arc which the character travels. So, the PLOT of Jurassic Park is that an accident at a theme park prompts the phoning of lawyers and an inspection at the island, which pulls in our main character and his then girlfriend by way of their profession. The dinosaurs escaping then become the PROBLEM he has to solve or be a central part of solving, and in so doing it propels him towards his internal change (story)
3. Just because they aren't the same, doesn't mean they're not interlinked! One of these informs the other and the relationship is never simple or easy to break down. Monster movies rely heavily on EXTERNAL plot devices to create the need for characters to enter the story (Godzilla, Jurassic Park, Jaws) whereas some movies, are much more subtle, the characters decision making informs the plot, their internal issues can directly CREATE the plot, the antagonists and their problems.
So it all comes down to what kind of movie you're making! But no matter what it is, make sure that you start with a GREAT character who has issues that need resolving, and then your plot should begin almost to write itself as you put this fictional human up against a series of challenges to test them... and hopefully, if you've done it right, your fictional human learns something important in the process and isn't quite the same as the one you started with!
Try it out.
Does this method work for you? Let me know.