The Heads Up Display measures how well the page you’re on is driving traffic to the stories linked off it. The Dashboard is all about how your pages are performing regardless of how people got there. A page might be getting a ton of traffic from social media to make it the #1 in your Dashboard, but it will only be the #1 in the Heads Up Display on your homepage if it's getting a lot of clicks from this page.
If a story is performing well in the Real-Time Dashboard but not on your homepage, it may be an opportunity to optimize the link position, image, or headline on your homepage.