Even though I call your book, article, essay - whatever - your "story," there's more to it than that. "Story" actually refers to the emotional part of what you're writing. "Plot" is the physical part. If Timmy falls down the well and Lassie goes for help, that's a Plot. If Timmy falls down the well and Lassie, for love of her boy, goes for help, that's Plot and Story.
The Plot is the outline of your book, the structure; it defines what happens. Story is the heart of your book; it's what keeps the readers emotionally invested in the characters and makes them want to keep reading.
We've all read books where there's a lot of action. The hero is involved in shoot-outs, car chases, daring rescues. At the end of it, however, we set the book down with a feeling of dissatisfaction. There was something missing - but what was it? What was missing, most likely, was enough of a Story, the emotional aspect that makes the reader care what happens to the characters.
Then there's the book that's all heartbreak, angst, emotion. But nothing else much happens, because there's not a strong enough plot. Without the skeleton of a Plot to hold your creature up, it's all muscle and soft tissue; pretty to look at, perhaps, but unable to really go anywhere.
A good book should have equal parts (or close to it) Plot and Story, with one leading naturally to the other. Let's look at "Gone with the Wind" again. This is a very lightweight interpretation of a particular scene, but I think you'll get the idea.
Plot: Scarlett, a recent widow, helps out at the Confederate Ball, a fund raising event to support the war.
Story: She sulks, resentful of the fact that her widow status prevents her from joining in the fun.
Plot: Against all protocol, Rhett Butler bids on Scarlett for a dance.
Story: Despite the disapproval of the other ladies, she happily accepts.
Plot: As she takes to the dance floor, Scarlett, in her black widow's weeds, stands out from the other young ladies in their brightly colored ball gowns.
Story: Knowing she has incurred the wrath of nearly everyone there, Scarlett nonetheless joyfully bounces around on the dance floor with Rhett.
Put another way: Plot is what happens; Story is how your characters feel about it.
"A great book should leave you with many experiences, and slightly exhausted at the end. You live several lives while reading it." William Styron