You've described the main UX difference in your question: "Remember me" will present a pre-filled login form, but with "Stay signed in" the user will be straight into their account. Displaying the login form adds an extra step for the user, but ensures they know which account they are logging into and allows them to log in to a different account if they want to. Both methods allow any user who has control of the browser to access the saved account.
There are some more subtle security differences, but exactly how they apply will depend on how your login system works.
A secure login system will try to minimise where it stores and processes the password. For example, it might ask for your username and password, then return a session ID. The application internally knows who is logged into that session and that their password has been verified, so it doesn't need to check the password on every request and the password does not need to be stored by the users browser. If you implement a "Remember me" feature then you will need to store the user's password somewhere, probably in a browser cookie. Anyone who can access that cookie can then discover your password and use it on their own computer, and can provide it when asked to verify it for sensitive operations such as changing the password.
It is also good practise to log users out of other devices when they change their password, because they may be changing their password because they know it has been compromised and they want to ensure that anyone who knew their compromised password no longer has access to their account. If the password is verified for every request then that happens automatically, but if authentication relies on the session then it should be implemented as a security feature.