Not logging on with Sysdba Privilege

Hi All,

     Your wisdom is needed.  I have a user that has the sysdba privilege.  There was a time when the user could log in either way:

sqlplus user@test as sysdba

or

sqlplus user@test

I was away during the government shutdown and the password for this user was changed.  Every since the password change, the user can only log in as sysdba.

  I keep receiving the ORA-01017: invalid username/password; logon denied

Can anyone assist with this dilemma.  The user is tied to an application.  I have been told that this user needs to execute some things as sysdba, but also needs to execute some things with out the sysdba privilege.