Importance is just another piece of metadata. Similar to department, keywords, or content type; these metadata can be used in a card mapping rule -- if a piece of content (article, banner, or collection) matches the metadata filter for a card mapping rule, then the designated card portrays the content in the browse page.
Importance is one of the metadata types used in card mapping.
As Neil indicates, share an image or describe your intended layout and we might be able to help more.
And let's make sure you're using a custom preflight app (we used to call these Preview apps until last Friday) -- and you're clicking the Preflight button. If you're not using the preflight workflow, you may face 3-5 minute waits after clicking Publish before the new content is available in your app. This might be adding a layer of confusion to your process.
HTH;
Colin