Dance and writing are alike because there are so many styles, themes and techniques, and in the end it all comes down to execution. Yet, I don't think that dance is as much a good metaphor for writing as it is inspiration.
What does a good dance have that your writing needs?
Your muscles tighten in ways you don't even realise when you're captivated. Your body is transfixed, absorbing and calculating. Something that is "tight" in the sense that it is pulled together, not strained, is noticeable. There is a reason you can hear the word "tension" in "attention."
Partnered or group dances are fantastic because you can observe the interaction between characters. The best stories have impressionable and memorable characters. When they come up against one another, you can see how it impacts their personal and relationship developments.
Dance uses music for mood, lyrics for dialogue. It is "show don't tell," subtext, and plenty of action. It can accomplish everything you can in words with none. We take so much from what we observe and bring it into our art - whatever the medium - to tell a story. It's that kind of powerful core you need to have in your fiction.
When you are captivated by something, it stirs inside you. There is as much a beat in writing as their is in music or dance. Rhythm is the pacing, the steps underlying your story, breaking out onto the page for the reader to feel the vibrations.
All the quick steps and rush of the music can cause a stumble. Jerky or clunky movements subtract from the magic and leave an audience braced for disaster. A dance where two characters conflict can hold tension without losing fluidity. The same can be done with writing. Rereading, rewriting, and editing your work allows you to concentrate on mastering a finesse which is often absent in a zealous first draft.
Writing is not a simple endeavour, but with passion and perseverance it is an immensely satisfying one. Treat your writing like a dance, and with a lot of hard work you'll succeed.