This has been bothering me all the time, but I was too lazy to find why this happening. When you come to a site that prompts you for a username and password (such as SharePoint portals for example) you prompted for password every time, regardless of "Save password" checkbox.
The issue and solution is actually easy. IE only logs you in automatically when in intranet (e.g. browsing locally). And for Internet, you will always be asked. To override this, you should select "Automatic logon..." in security custom options. See screenshot below.

For more information, check this KB: KB258063
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