Summary of existing answers describing differences:
Provide an additional means to organize messages that behave differently independently from, but also in relation to, tabs and categories
Specific differences between importance and starred in how they operate, how much control you have in their usage, what they effect, and when they do so.
importance = ...
A little bit late, but since I found this as the first result when searching for this very issue, I should add that it's possible.
Set up the priority inboxes and your labels, if they aren't already.
Apply a filter to archive "work" messages coming in so they skip the main inbox.
Go to Settings > Inbox > Options next to the inbox section you want ...
You can’t control if a message will be marked as important in the recipient’s inbox. The priority marks in Gmail are algorithm-determined so you can’t control them.
How importance ranking works
To predict which of your incoming messages are important, Gmail automatically takes into account a number of signals, including:
Who you email: If you ...
It doesn't look like you can. When configuring a custom section for Priority Inbox, your option for messages is either:
Inbox and one of (Important and Unread, Important, Unread, Starred)
All from one particular label.
You can't define a search (label:my-label is:unread for instance) for a priority inbox section.
The use of the priority inbox is controlled by the settings:
To turn it off:
go to : gear, settings.
Click on the inbox tab.
Change inbox to default.
click save changes.
Note: I also have don't use markers/don't use my past actions/don't override filters selected.
This is feature is not currently available. You can suggest new features (and vote for the most common suggestions) here: http://mail.google.com/support/bin/request.py?contact_type=suggest (perhaps using the "I have another idea" option at the bottom)
As an alternative, have you tried the Multiple Inboxes lab feature?
Click the gear icon -> Settings -> Labs, and find Multiple Inboxes.
When you have activated it, a new tab appears on the settings screen.
It lets you divide your inbox view into multiple sections. Each section displays the results of a pre-defined search.
It should be possible to specify ...
Google says that label filters will never match incoming mail but that's a lie. label:important works fine for me. label:whatever seems to work fine too, if you created another filter that assigns that label. Perhaps what they were trying to say is what they say clearly anyway in the 2nd page of the filter's definition: the filter works on incoming mail only;...