I'd been having this problem on and off - randomly my iTunes login would just stop working - I'd get a login prompt but couldn't log in. Until I read this discussion thread it had never occurred to me that how I used SAFARI would affect whether ITUNES works, but it does. And no special plug-ins are required.
In order to totally break iTunes login on a mac, just do this:
(1) Exit from iTunes and launch Safari
(2) In the Safari menu. turn on "Private Browsing"
(3) Launch Preferences and navigate to the Privacy tab
(4) On the Privacy tab, select "Block Cookies:Always" and then click "Remove All Website Data" (and confirm by clicking "remove now")
(5) Launch iTunes again
RESULT=> any effort to sign into your iTunes account will fail silently. If you have the "automatic downloads" feature turned on, you'll get a "sign in to enable automatic downloads" prompt that doesn't work. If you try to "download all available updates" for your iPhone/iPad apps, you'll get a "log into your account" prompt that will fail to work. There is no warning message that explains why login fails, but the reason why it fails is that iTunes wants to set an Apple.com cookie in Safari. Since you deleted cookies and aren't allowing new ones in, iTunes can't log in.
I must have changed my iTunes password a dozen times before I figured this out.
To fix it, turn off the above-average secrecy settings in Safari. In particular, set your "block cookies" to either "never" or "from third parties and advertisers". And don't delete the Apple.com cookie once it's been set.