Hi
First of All
Backup control file is a binary backup of control file which can be read and interpreted by Oracle.
Backup COntrol file to trace is the backup of control file in ASCII (text) mode so that one can edit the file to recreate the control file. (eg u can edit the control file to add/drop some parmeter, file, etc)

Now, there is SCN # (system Change Number) generated by Oracle and written simultaneously to Control files, data files and Archived log files. When taking cold backup, the DB be shutdown (Shuldnot be Abort) and Physical copy of all the files. control files, data files, etc
SO when one wants to restart the db with the backed up files, all the files (including datafiles have to be copied. Merely copying control files/RL files will not serve the purpose, as when DB starts, it cheks the SCN# in all the files and shud match. so all the files need to be copied
(When u copy the file here, it shud be the binary control file and the trace file)
feel free to ask if any further doubts