You need to tell Firefox to remember your usernames and passwords. Select Tools > Options > Security. Ensure that "Remember passwords for sites" is checked. For more information, read this article: Remembering Passwords. After changing the settings, if Firefox is still not remembering your usernames/passwords, check here.

It is NOT a matter of remembering of password. It is what it does NOT do after log- in is successful!!

Sorry, I still do not understand your question completely. You say that log-in is successful in both Internet Explorer and Firefox. But in Internet Explorer, it shows a menu list. What exactly is this menu list? It would be helpful if you provide a screenshot so I can understand your query better. Or, can you give an example of such a site?

Menu List is List of links to proceed further into the website. The problem is as you have exactly understood and the distinction is between the use of firefox/ Microsoft Internet Explorer browser. Sorry I do not know how to create a screenshot- sorry.