As I understand it, "Remember Me" will retain your login details making future logins simpler and quicker while "Stay Signed In" simply prevents any automated logout.
Which one you prefer depends on the level of security you wish to maintain or that is inherently present on the technology you are using. On a smartphone, for instance, the user may lock their phone with their own security measures so you don't necessarily need to worry about logging the user out - their inherent security measure effectively replace yours (to a degree). However, if the user is working at a public terminal, you should not only worry about making sure that they are logged out when they leave but also reducing the session length to minimise the amount of accounts that are accidentally left logged in without their users present.
These examples are, of course, fairly extreme and you would need to use research and judgement to decide the best measures for your situation.