A storyboard allows a movie director to communicate his vision of the scenes within the movie to other personnel involved in its production. For example, the wardrobe department can better grasp what styles of clothing the characters should wear. Location scouts receive an enhanced feel for what buildings and landscapes to seek out, while the prop master knows what props to place in each scene. Basically, all crew members can visualize the movie's desired look, and better prepare to make it a reality.
With a storyboard, a director better visualizes the look of each shot she needs for the movie. This way, she can decide when to use close-ups, and when to use wider shots to best tell the story. Looking at the action occurring inside particular scenes, a director figures out ahead of time where she wants the camera placed and where she wants the actors positioned. This avoids waiting to make these decisions when everyone is waiting on the set, and precious time is steadily ticking away.
Just because a movie screenplay is written with certain scenes, that does not mean it will be filmed that same way. Some scenes might simply not work as written and might need strengthening -- or scenes might not work at all. A storyboard allows you to visualize if certain films work in the order presently arranged, or if those scenes work better in a different order. A storyboard allows you to take out those scenes that don't work ahead of time, and install more powerful, interesting scenes, before creative efforts and manpower are wasted in preparing and shooting ineffective scenes.
A storyboard is just like watching a full movie, just in a different form. Through it, you get an idea of the story's overall feel, as well as its pacing. The storyboard encourages creative assessment and discussion about any points or setups within the story that don't add up or that need tweaking. Because you can add dialogue and effects to digital storyboards, judge the dialogue strength and whether the story contains all of the right elements before the cameras roll.