✔ Remember username (Email or Account ID)
Does not work when browsing in 'Private' mode.
Not recommended on any shared or public device, only select this option on a trusted device!
Your password still remains required to ensure authorised account access only.

✔ Stay signed in - Use only on trusted devices!
Does not work when browsing in 'Private' mode.
A secure session cookie is stored on this device instead of a password, permitting instant access for up to 90 days of last sign in on this device, or until cookies are cleared any earlier.
When you're signed in to your account from any device, under 'Devices' you can remotely remove any of your 'Stay signed in' devices anytime.