Storytelling would help improve your writing skills because of two things: Hearing what you're saying, and instant feedback from your audience.
In terms of hearing what you're saying, I think we've all had moments where something that sounded not just well-composed but absolutely masterful in your head ends up sounding contrived or silly when you say it out loud. The disconnect between thinking the words and hearing your voice say them brings clarity and a new perspective in a way that I've found writing doesn't; telling stories will help you develop the skill of analysing your writing in this way without reading every passage out loud - something which, if I'm honest, I still sometimes find necessary.
The biggest benefit I can see, though, is the fact that when telling a story, you can gauge your audience's reaction instantly, not just to the plot, but to smaller things like characters and even words:
It will help your narrative structure; you will quickly learn the degree of tension and progression required to keep the attention of your audience, and, as <PERSON> said, you will learn a lot about the order and composition of narrative elements.
You didn't mention it in your question, but you will also learn to create compelling characters. Speaking for a character engages part of your brain that simply isn't normally active when you're merely writing about them, a part that is responsible for understanding motive, emotion and growth, which are cornerstones of strong characters. 846
At the same time, you will learn to present only those parts of a character which are necessary for developing their role in the story; unnecessary cruft only weighs down your character if it serves no purpose for the reader.
'user2738698' seems to want to imply that storytelling won't help your word usage, but I disagree. He's totally correct to say that writing a story as you tell it would sound corny; the written word just lacks the subtleties of emphasis and intonation available in the spoken word. But telling stories will teach you the effect a single well-placed word can have on your audience, and after the first few times you've struggled to explain what exactly you meant to a misunderstanding listener, you'll learn the importance of choosing precisely the right word to say what you want to say.
I could go on about the general helpfulness of immediate feedback, but I think you get the idea. The last thing I'd like to mention is that storytelling will teach you to tailor your diction, grammar and plot to your target audience; just don't get stuck in the mindset of a person writing for young children - unless you want that to be your career - just because they're the most likely age group to sit down and listen to a story. 873