"Remember me" feature is used to preserve the login name and password entered by the user. Next time the user will visit the login page, the form can be populated with the info, but this not implies the user will still be logged in.

"Stay Signed in" will keep the user logged for a longer period of time.

Practically, the two functions can be identical because usually it is used as a cookie for the period of time the user stays logged in, it is not safe to be unlimited, but theoretically, the difference is if the user remains logged in or just has the form pre-filled with the info and he just need to press the submit button.

"Remember me" feature is usually used to preserve the username entered by the user. Next time the user will visit the login page, the form can be populated with the info. (Update: After a better research, I read some suggestion to improve the UX for this, like changing "Remember me" with "Remember username on this computer" - https://www.sitepoint.com/3-rules-painless-account-ux-login-screens/, Do we really need a "Remember me" option on a Login page?). In this case, after you close the browser, you will be logged out and after you will open the login page, the username will be pre-filled.

"Stay Signed in" will keep the user logged for a longer period of time, even you close and reopen the page, using a cookie.