Specified cast is not valid. in sql code example

Example: specified cast is not valid. (sqlmanagerui)

The GUI can be fickle at times. The error you got when using T-SQL is because you're trying to overwrite an existing database, but did not specify to overwrite/replace the existing database. The following might work:

Use Master
Go
RESTORE DATABASE Publications
  FROM DISK = 'C:\Publications_backup_2012_10_15_010004_5648316.bak'
  WITH 
    MOVE 'Publications' TO 'C:\Program Files\Microsoft SQL Server\MSSQL10_50.SQLEXPRESS2008R2\MSSQL\DATA\Publications.mdf',--adjust path
    MOVE 'Publications_log' TO 'C:\Program Files\Microsoft SQL Server\MSSQL10_50.SQLEXPRESS2008R2\MSSQL\DATA\Publications.ldf'
, REPLACE -- Add REPLACE to specify the existing database which should be overwritten.