The password isn't saved by some function of Hotmail. It's a function of the settings on the computer. If you're accessing Hotmail from a public computer (Internet cafe, library, etc.), it's probably designed to delete temporary files and cookies on a regular interval, as you'd want it to. As for not working on your personal computer, are you running CCleaner or any other cleaners or privacy protection-type applications? These are almost always the cause of this type of behavior.
A creation of a password issue (password was not long enough) for a customer. The problem is the duration of the password. This is controlled by the local policies on the computer.
Here's the solution:
Run>gpedit.msc>Computer Configuration> Windows Settings> Account Policies> Password Policy> Minimum Password Age should equal 0. Mine was set for 42 which is why it worked for a while and then stopped. I have rebooted my machine twice and the fix has held.