If a user is login with Social LOGIN then social button should disappear after login

I am using CBConnect 4.0.2 in Community Builder 1.9 with Joomla 1.5.23.

If a User uses Twitter / Facebook / LinkedIn to log into Front Page then after login , social buttons should diaappear. See pic here, a user is already login while social buttons are still there.

That's Linking. It allows them to link a social site account to an existing CB account. They're only present if the user isn't linked already. So they must have logged in using normal CB login instead of a social site login button. You can disable linking within CB Connect > Config if you don't want it.

If Linking is disabled then does this mean that when a User login with his Social login i.e Twitter account then his Twitter details will not be transferred or Registered in CB User List as a User.

No, Linking is strictly to link an existing CB account to a social site. They did not sign in with Facebook they're seeing a Facebook link button as once the accounts are linked the button has no purpose and won't display.

If a User Login with CB Username and then later links to Facebook then this Button will disappear and other Buttons will remain there.
Why other Buttons will remain there, why option is given to him also link to Twitter account. This action will not bring duplicate data of that User ; and further that he may have different name & data on Twitter account. How this additional information will be stored in CB User list.

Am I missing the understanding of 'Linking an existing CB account to a social site'; what actually happens in this in terms of storing the User Data ?

Linking does not synchronize data. It just links their social site ID with their CB account. This means if I logout I can then use that social sites button to login. This also authorizes your social site application to make API calls on that users behalf. What you need is entirely up to you as I said you can turn linking off if you want.