I can confirm that Password Protected pages still work as normal in 3.4.1.
The password protect form does submit the password to /wp-login.php?action=postpass where that file deals with authentication. Upon successful or failed login the file will redirect you back to the page the user was on.
One thing to be aware of is that the login file requires the browsers Referer header to be set so it knows where to send the user back to. If your browser doesn’t send this header then you will encounter the issues you are having. You may have it disabled in your browser through some browser extension or being in private browsing mode.
Now possibly in the past version of WP the post/page password form used to send a referer field but it doesn’t in the latest version. I will investigate further.
UPDATE
I’ve tested on an older version of WordPress (3.0.0) and the behaviour is the same as the current version. I’m going to stick with the idea of your browser no longer sending the referer header as the cause of your issue.