When a user enters the password at the password protection screen, there is an option on that screen labeled "Log me in automatically each visit", which is enabled by default. That option saves an encrypted version of the password in the user's browser. That is why the next time the user visits the forum they will not be asked for the password.

A user who is signed in to your forum keeps their access once they enter the password, even after you change it. A visitor who isn't signed in is asked for the password again if you change it, if they use a different computer or browser, or if they reset their browser cookies.