There is an annoying problem experieced by Apple Mail users with GMail accounts.
GMail creates a virtual mailbox called “Important”, which displays your seemingly important mails. That’s all good and fine until you attempt to access GMail over Apple Mail application on your Mac.
What happens is, the e-mails in your “Important” folder don’t show up in the inbox of Apple Mail. You see them via GMail Web, but not in Apple Mail. Considering that those are actually important E-Mails, the problem is pretty frustrating.
I have constructed a workaround for this annoying problem, which might help others out there.
First; go to your GMail settings, go to the tab “Labels” and ensure that your “Important” folder is enabled via IMAP.
After this step, Apple Mail will start showing up your “Important” folder. You can see the mails by accessing that separate folder.
However, the “Important” mails still won’t show up in your inbox. To work around this problem; go to the menu Mailbox – New Smart Mailbox and include your inbox + the “Important” folder.
Voila! From now on; you will click the smart mailbox to see all of your E-Mails.
If there is a direct way to merge all mails into the inbox, please let us know in the comments.
Dr. Kerem Koseoglu 
I could hug you!! I’ve been dealing with this for almost a month now!! One day I noticed that an “unread” number was still showing on the mail icon which I thought was strange since I just checked all the mail in my inbox (I have several emails, including 2 gmails). Then I noticed a ton of emails in the “Important” folder that never showed up in my “Inbox”. I wonder how many truly important emails I have missed because of this!?
Anyway, this was the only thing I found that actually solved the problem so thank you very much!!
I’m glad I could help! Let’s see if Apple / Google provides a real solution instead of a workaround someday.